I am very interested in the narrative element of D&D.  One of the overlooked bits I see is players and DMs being interested in whether a weapon or suit of armor is average, fine quality, superior or exquisite craftsmanship.  In the narrative, it only seems to matter when describing an NPC so that if he is wearing fine quality items, you think he is more affluent and has a higher status.  But otherwise it doesn't matter.  I have a proposal to make it matter, and I want to make sure this doesn't break the game.  I don't think this does.

An average quality item (weapon) does the regular PHB damage.  A longsword does 1d8 plus modifiers for damage.

But a Fine Crafted Longsword would do 1d8 damage, but the player also rolls a d6 and divides it by 2.  If the d6 divided by 2 is a higher number than the roll of the d8, then the weapon does the d6 divided by 2 damage instead.

A Superior Crafted Longsword would do 1d8 or 1d4 damage.  And an Exquisite Crafted Longsword would do 1d8 or 1d10 divided by 2.

So let's look at the real difference ...

1d8 raw = 4.5 pts damage     Average Weapon

1d8 or 1d6 divided by 2 = 4.67 damage     Fine Crafter Weapon

1d8 or 1d4 = 4.81 damage     Superior Crafted Weapon

1d8 or 1d10 divided by 2 = 4.85 damage     Exquisitely Crafted Weapon

The interesting scenario might be what happens when you are using a dagger or some weapon that only does 1d4 base damage.

And to keep this in perspective a magic weapon or other +1 weapon would yield 5.5 damage over the base 1d8 weapon.  Therefore I don't think this breaks anything except possibly the DMs need to take notes.  But by adding a mechanical aspect to the description of a high-quality weapon might be worth it.

Another way to think about it is to maybe give the finer crafted weapons a "minimum damage"
So like a "Fine Longsword" would be like 1d8(min2) and a superior could be like 1d8(min3). I feel like the d6 method would become cumbersome to keep track of and players would forget they had it. But the Min-damage method is like giving a "kind-of+1" without increasing the maximum damage or accuracy

I also like the idea of different qualities of equipment but I think your approach is waaay too complicated and adds a lot of unnecessary burden on players and the DM.

The easiest way to do this is to just say a 'fine' weapon is a +1, 'superior' is a +2, etc. It's not unreasonable for nobles and powerful people to have access to these items.

If you don't like the fact that the weapons are magical or think the +1 to hit is too strong (which is fair). Just give it a +1 to damage.

You don't need an incredibly complex system which doesn't work with d4 weapons to emulate weapons being of higher quality.

Remember K.I.S.S. (Keep It Simple, Stupid). The simpler a system can be, the better it is.

imo, you should redo your categories and go down, not up, with the idea that only very fine weapons would be eligible for being enchanted in the first place.  So a magic very fine weapon would start at +1 (so sure, a superfine non-magical weapon might do a wee bit more damage than a fine weapon, but not enough to round up to +1).  That means a standard PHB weapon is already fine or very fine.   Go down, not up.  so you'd have a crappy penalty and a very crappy penalty. ...but no extra damage for super sweet, but nonmagical weapons.  they might be more expensive because they have fancy gems or polish on them, but they don't actually do extra damage.   someone shoves a dagger between your ribs, it really doesn't matter how nice it is, it just needs to be skinny and pointy.

and to tepil's point, you're overcomplicating it imo. to do your approach, imo, you'd need to also scale up magical weapon damage. a non-magical weapon shouldn't do more damage than a magical weapon in almost any scenario....all imo (again because those magic weapons are already superfancy to begin with).

 also, if you feel like spending money, look at this pub, it's already done what you're trying to do (and moreso)...which is a wholesale replacement of the PHB weapon system (not that i recommend it as it certainly isn't simple - but with the sales and ratings, its obviously on to something).  and I definitely wouldn't recommend it if you have an older computer, the pdf file size is huge and chock full of art that inflates the page count, which breaks my PC when trying to read it.