Does Xbox Series X|S controller work with 14500 Lithium-ion rechargeable batteries?

No it does not. The Controller is designed to use AA batteries, so you will need standard rechargeable or disposable AAs, or the Play and Charge kit.

I'm not able to find any references on what the specific dc to dc step down regulator is inside the controller.

Also, I looked at many Play and Charge kits and compatible devices. All of them are essentially two repackaged NiMH or Ni–MH batteries.

I'll get some rechargeable NiMH batteries and use those.

[quote user="KI11RB33"]Unnecessary...instead of taking the rechargeable lithium ion battery and just putting it in the controller,[/quote]Where you cannot replace it when it runs out of charge and carry on...

they sell it separately as an accessory.

Which
you have the choice not to buy.

rechargeable controllers are last gen...

And they sucked then, too.

The turtle beach xp500 and xp510 are similar for the Xbox 360 you just use the Bluetooth attach it to the controller. It would be great if turtle beach made a completely wireless headset which didn't need any connection whatsoever (game sound and chat) to
the controller and which will work on both 360 and 1. I have the turtle beach x42. The reason why I don't like the xp510 is that it has a rechargeable lithium ion battery instead of being able to use rechargeable energizer batteries or normal AA batteries.
Lithium ion batteries aren't very reliable. My PC runs on a 6 cell battery is 3 years old (Windows 7 intel core i5) and now the battery depletes rapidly lasts half as long as when I brought the PC new (unless you like trying to source spear parts from turtle
beach this can be a bit challenging especially if you don't live in the US/UK but instead live in NZ or Australia).

It is a Lithium Ion Battery with a capacity of around 1400 mAh