People have had controllers with built in batteries since the PS3. We strongly dislike them for good reasons. My old 360 controllers are basically like brand new. My Sixaxis and Dualshock 3 controllers on the other hand are basically hard wired controllers at this point given how short the battery life is. I could replace the internal batteries, but even then in 3 or 4 months when I finally get around to using the controllers the batteries will have fully discharged and I'll have to recharge the controllers before use or play hardwired while they charge. Even when they were new, my wife and kids would discharge the batteries while I was sleeping or at work and I ended up playing hard wired as much as I played wirelessly. Meanwhile my 360/XB1 controllers are mere seconds away from having a fresh charge for any reasonable number of controllers which makes them fantastic when I'm playing with the kids or have company.
Your problems aren't the fault of the rechargeable battery. Going months without use/charge shouldn't yield any other result other than a dead device. That's like deciding to put my phone in my book bag for a week and hoping its still charged when I decide to use it again. You also reference 10+ year old controllers as your example, but rechargeable batteries have improved since, so its not fair to compare them. However, I don't have any kids of my own, so I'm sure it can be a headache if your kids are constantly killing your controllers battery life (I wouldn't allow my kid to use my $150 controller to begin with (I let my nephew play with my standard xbox controller.... I'm a bad uncle)). I hope I'm not coming off as a jerk or anything, but I have various devices around me with non removable batteries, some ranging from 5+ years old, and they all work perfectly fine. Again, I understand there can be a bad batch here and there, but generally they work as intended for many years. On a similar note, do you own a switch? Its literally 3 built in rechargeable batteries.
all of this is meaningless in the end, these manufactures are able to produce these devices cheaper than ever, but charge more than ever for them. So if your device breaks out of warranty, too bad, go fork over another $$$ for a new controller (
and yes, that sucks a lot).