Before anyone says no, I propose an ability or high-level Artificer spell that has a long cooldown.  

The ability would go...  

Arcane Engrainment:

At level 17, you can sacrifice all of you 4th and 5th level spell slots in order to make an object you have infused permanent.  This ability MUST be done when you have all of the required spell slots of 4th and 5th levels unspent (for example, immediately after a long rest.)  This ability becomes available again after 1d4 days per level of infusion made permanent.  The item no longer counts against the number of infusions your character has and retains its magical properties if your character dies.

This would give the artificer some more bite and let them be the Mastercrafters of Myth and legend.

Why would they need a class feature to do that?

Magical item crafting is a thing that has rules in the book.  They're bad rules because Wizards can't figure out whether it wants players to be able to make/buy items or only find them in the Dankest Recesses of the Earth, but they're there.  Any DM with their head screwed on straight should be able to figure out that an artificer wanting to make a permanent version of one of their infused items should at least get a time discount - the class outright states "these are prototypes for permanent items", so the artificer is already doing their R&D while they're monkeying with the infused item.  If they're an Artillerist or Battlesmith and get an additional discount on wands or armor, making a permanent version of their infused wand or infused armor should be dirt simple.

Making a permanent magical item has monetary and time costs.  An artificer with their infused item to act as a prototype and proof-of-concept should easily be able to slash that time investment pretty heavily, and might get a slight discount on resource cost as well depending on the game, but they'd still use the crafting rules in the book.  I'd probably rule that an artificer who's trying to create a permanent version of one of their infused items can do it in half the time, as the infused item serves as a model they can use to avoid time-wasting trial-and-error.  The resource cost would be harder to cut into since magic items need stuff to be permanent, but even then I'd likely give them at least ten percent off.  More if they could make a good case, or successfully completed a side adventure to secure some of their own materials.

This is the sort of thing a DM exists to rule on.  There's no need to clutter the class with a "can craft magical items!" ability, especially one that amounts to "can make a new permanent magical item worth thousands of gold once per day on any day that doesn't require adventuring".  The only reason anyone would need a rule like this is if they're playing AL, and if you're playing AL you should stop doing that and find a way to try and play actual D&D instead.

The reason is because I'm also trying to think of "archetype-specific" infusions for the artificer.  There are a few infusions I think would go well with each archetype, but the problem is there is no set rule for making infusions permanent.  It feels like the infusions are "on loan" and it's a real non-plus for the whole class.  

Homebrew should be for more character flavor, in my book.  The artificer could really use a boost like this, because the battle smith and the alchemist are pretty much balanced enough to be solid official classes.  The class itself just needs to be nailed down a bit more, because I want the current artificer to become official, but Wizards most likely won't put its stamp on it unless it feels solid enough like an actual magical crafting class.  

The infusion system is a direct result of people complaining that artificer class features from the original Alchemist/Gunsmith UA amounted to magical items that could be lost, stolen, or in some cases used up/consumed and leaving you with nothing at all for your efforts.  So instead they give you a class feature that says "you have the ability to hijinks magical items into existence, so if you lose it you just replace it."  It's a better system because it allows artificers to be more flexible with their magic item usage and have infusions for situational items on standby, but the class as a whole is still Wizards trying to balance the fact that they want magic items to be special, rarely-acquired rewards in most official games/modules with the fact that artificers eat, sleep, live and breathe magical items and magic tinkering.

Infusions don't need to be permanent and it's barely even homebrew to state that artificers get special perks for crafting magic items; it's just in-game logic.  Who's going to produce a Bag of Holding faster/more effectively - somebody working off of a basic idea they got from an old textbook once, or somebody who's got a Bag of Holding he can examine and who has made a prototype Bag of Holding every day for the last six months?

I think increasing the number of items infused and infusions known would be a simpler fix.

I think it ought to be something similar to the crafting system where a character proficient in smiths tools can spend gold and downtime to make a sword, an artifice should be able to do the same thing to enchant said sword without it counting against his total infusions.

I would argue that being able to Replicate a particular item should allow some other process to actually craft it into the real thing at a lower cost than doing it from scratch.  But that would inherently require a rewrite of the crafting rules for 5e, since they are holdovers from a much slower playing era and do not fit with the current state of the game.

I can understand where some of the complaints are coming from, but honestly its up to your DM and your game. 
I would say you do t need to make it a class feature, but that system of “crafting” could just be how your DM or what have you, decides your Artificer creates their magical items. 
i would say make it less of a clear cut class feature and make it more of Artificer specific crafting rules that scale through the levels and allow you to create the magic items you want. 
On top of that, most magical items that artificeres get as infusions are just twisted versions of something that already exists in the DMG or somewhere similar so most of the time you would be better of just making a version of the original 

It's not even a house rule that Artificers can make Common and Uncommon magic items at half price and quarter time.

According to Xanthar's, all you need to make a permanent magic item is the right proficiencies, time, money, an exotic ingredient that is up to the DM, and the proper formula to make that specific magic item. And since nowhere in any of the books does it mention how to acquire these formulas, my DM said that my Artificer knows how to make everything he knows how to make.

The only real house ruling you need to do is decide what rarity each of the Artificer-unique items are.

I’ve added a homebrew rule for Artificers infusions 

the max no if items that can be infused is 1/2 the class level (min 2)

and if a level 4 artificer takes 4 in game weeks to cast the same infusion on the same item every single consecutive day the infusion becomes permanent

this does mean that no other infusions can be placed on a different item during this period and if a day is missed the permanence consecutive day counter resets 

Magic Item Adept

When you reach 10th level, you achieve a profound understanding of how to use and make magic items:

• You can attune to up to four magic items at once.
• If you craft a magic item with a rarity of common or uncommon, it takes you a quarter of the normal time, and it costs you half as much of the usual gold.

Most of it is already there. The only adjustment i would make is, to also give a bonus to rare and legendary items at higher lvls.

Quote from Bhuraelea >>

Magic Item Adept

When you reach 10th level, you achieve a profound understanding of how to use and make magic items:

• You can attune to up to four magic items at once.
• If you craft a magic item with a rarity of common or uncommon, it takes you a quarter of the normal time, and it costs you half as much of the usual gold.

 

Most of it is already there. The only adjustment i would make is, to also give a bonus to rare and legendary items at higher lvls.

 But artificers can't normally create duplicates of rare or legendary items, so why would they be able to manufacture them any easier?

Artificers are as strong as they need to be. Money is no object at high levels, and my high level artificer could basically hand out any uncommon items the party desired. Having access to permanent rare+ items on demand is absolutely going to break some games and will just lead to more DMs banning Artificer outright.

If you want to let them craft really good things, houserule it. It should not be official because that reduces the overall appeal of the class, aside from the fact that it would be a huge power boost to a class that is already quite strong.

When you look at these things, you need to think whether it would make the class better for everyone, not just better for your game. This is a case where one person's "wouldn't it be cool..." is another person's "no way in hell."

I wouldn’t say handing out. 1/4 of the time needed is still 500 days for a legendary and over a month for a very rare item. 

Quote from Bhuraelea >>

I wouldn’t say handing out. 1/4 of the time needed is still 500 days for a legendary and over a month for a very rare item. 

 Legendary are called that for a reason - they are one of a kind, and usually have a strong history associated with them.

Depending on the style of your campaign, everybody could just do downtime stuff for a month without compromising anything in the campaign.

Gold... Down Time... Supplies...   these are for the privileged...   My Character has traveled from the freezing north... threw towns that had their shops burnt and vandalized...empty fields and dark forests...  sleeping in dugouts and trees...  what good is Gold except for gold plating...  As an artificer that has Tools and knowledge that can make clockwork items over night but cant even infuse a rock to glow permanent... My Character has to Gather supplies. Tinker and fail and tinker again.  My Character and His Teammates cant even combining their Power Infuse one Item...   "unless the DM says so"....

It seems to me that the point to using artificer infusions in enchantment is that they would be more efficient than using spell slots since they already have the complete "idea" of the item in their form. So it would still require time and money, but as mentioned the idea of formulas for magic items are pretty vague...