I imagine they didn't do this because they don't want to risk having the players create armies of extraplanar creatures.

Problem there is, what cleric will help someone capture a celestial

There are clerics of evil deities who don't care much for celestials. I can easily see a cleric of Tiamat, Asmodeus, Maglubiyet, or Lolth doing that.

If you are looking to solo trap the elemental from the Conjure Elemental spell without wanting to concentrate for an hour then remember to up-cast Magic Circle. If you don't then there's a whole minute where the elemental isn't bound by the magic circle due to the casting time needed to finish casting Conjure Elemental spell.

If your party is overflowing with cash/gems then there's no need to worry about using other spells to give you a better chance of binding the elemental - it disappears after an hour if it manages a CHA SAV. Just retry the next day. For those min/maxers or financially strained adventurer - hopefully the caster is a Divination Wizard with a low roll on portent or the party has a caster capable of casting bane (just tell the elemental to accept the effects of bane) to give an extra 1d4 lower roll on the CHA save. 

*note* technically the DM can pull a dick move and have the elemental disappear after the 1 hour cast since there's a grey area between the end of Conjure Elemental and the finish casting of the Planar Binding spell.
**Sage Advice** JC mentioned that "he'd allow" for the Conjure Elemental + Planar Binding to work in his games.

Cost calculation:

Magic Circle: *only if not wanting to concentrate for an hour* 100gp worth of materials which is the equivalent of 4 castings of ceremony spell to create enough holy water - alternatively buy holy waters, silver and iron
Conjure Elemental: Covered by the component pouch if bought
Planar Binding: A jewel worth 1,000 GP - can be created by combining jewel of the same type via the Fabricate spell. *should be cheaper if using IRL logic (larger the gem the higher the worth) - again up to the DM. There's also some argument about raising the price of the gem by being proficient with jeweler's kit through giving the gem the "perfect" cut.
Total of 1,100 GP worth of materials, 2x 5th level and 1x 4th level spell slot (remember to up-cast magic circle, plus time/effort per binding attempt. Up-cast this spell to get the best bang for your buck.

Can you communicate with the creature telepathically, or do you have to say your commands verbally?

Combo this spell with Summon Fey, Celestial, or Shadowspawn from Tasha's and you will have a great day! They won't resist the binding

All the "Summon [Thing]" Tasha's Spells last exactly one hour's time, so, they'd vanish exactly 6 seconds before this was finished, and even if not castinf any apell for over one Action steals Concentration...

...unless one caster does this spell and another does the summon. Combo Casting gets you those real long summon times. (Oh, and they gotta Extended Spell Metamagic to make it last long enough as well, since target needs to be in range for whole casting.)

Quote from NavySpheals >>

Combo this spell with Summon Fey, Celestial, or Shadowspawn from Tasha's and you will have a great day! They won't resist the binding

 Since Concentration's stolen by longer cast times, gonna need two casters at minimum.

What happens if yoh cast Planar Binding on a creature already being affected by another caster's Planar Binding?

Only the longer than action casting time requires Concentration.

Once casting is complete, no more concentesting!

You could always ask your DM if it feels appropriate to change it into a ritual spell.

heehehehehehheheheheheheheheheheheheheheheheheheh

At some point the control ends...Lets hope you have a plan for that.

A. a jewel worth at least 1,000 gp, which the spell consumes at the completion of the casting.

B. a jewel worth at least 1,000 gp, which the spell consumes on a failed save

RAW seems to suggest A.  Just wondering official rule vs DMs' personal interpretations.  Perhaps this was always intended as an expensive gamble (particularly for a low level party).

Instead of other people, bane and or feeblemind can be stored in glyph of warding.S

So can Planar Binding for that matter.

Not sure if the spell is worth the 1000gp gem for a day. At higher levels definitely, though.

It’s fun to summon an elemental with Conjure Elemental, then slap on this so you can keep them around for longer and without concentration.

If you start casting this immediately after you finish casting Conjure Elemental, you’ll have just enough time to complete it before the elemental disappears. Don’t forget your Magic Circle though!

Also if you’re a level 14+ Cronurgy Wizard, you can force whatever you’re binding to fail the save, which is helpful for resource conservation.

IMO it really becomes worth it once you hit 6th level spell slots, since then you can bind them for far longer. Casting this at 7th level or above can easily give you an extra ally for the rest of the campaign, as long as you’re carful not to let them die.

Also could be extra funny with an Invisible Stalker, considering they’re, well, invisible. Though their raw damage output is lower than an Air Elemental, and they don’t have Whirlwind.

Edit:

I now realize that Conjure Elemental specifies that the elemental becomes hostile when concentration is broken. Charm Monster can help with that.

Quote from DIemortalscum >>

How to capture and bind an empyrean

Step one: find the name of one using legend lore

Step two: cast gate and speak it name so they are summoned inside a circle.

Step three: get a cleric to cast bane and a wzard to cast feeblemind.

Congrats you now have a child of god to do your bidding.

Except Empyreans can Plane Shift and have Legendary Resistance, so it can literally just nope right out of your circle regardless of what it can roll for a save. Even if you have someone readying Feeblemind, it gets three LR's so it can just shrug off the Feeblemind first. Also deities have fiat to just nix attempts to open Gates to their planes, so you better hope your target's parent has their omniscience turned off. I'm not saying there's no possible way to bind an Empyrean, but given the stats this approach would need the complete cooperation of your DM to fly. More realistically, you'd have to go find one somewhere and beat it down first, then attempt to bind it. Realistically, this spell isn't for calling up and binding Legendary Outsiders; they've pretty much all got multiple LRs and ways to bust out of a Magic Circle or otherwise give you a very bad time as you attempt to cast this.

Bear in mind that the more beefy the devil/fiend that you summon, the more likely it is for the spell to fail. 

Saving throws are frequently things that Fiends and Celestials have advantage on. That is why it really pays to cast an inverted magic circle at a higher level and have an extra hour or two to recast planar binding, if it fails the first/second time. And have a banishment spell handy just in case (i.e. : cleric to pull your butt out of the fire ...) 

with regards to the criticism of feeblemind:  "How's it going to "do your bidding" if it can't understand you? It still recognizes its friends, and you're definitely not one of those. A feebleminded creature can't even understand the instructions you give it with Planar Binding." --- Instructions given with planar binding can be given and changed any time during it's duration. 

You have misunderstood the point of the feeblemind here. Target's Charisma drops to 1 (-5). There is no need for it to understand you while you cast planar binding, it just makes a charisma saving throw at the end of the 1 hour casting with a negative 5 modifier. It is then bound to serve you for the duration of the planar binding spell. 

Greater restoration, Heal or Wish can then be cast a day later and end the feeblemind and it is still bound to the caster, presuming planar binding was cast at a higher level. It might pay to cast a mind sliver cantrip again and again until it fails an intelligence save and rolls a 4 on the resulting modifier before attempting the 8th level feeblmind. Making sure that you have cast magic circle for a longer duration than 1 hour might be essential.