I turned off app auto-updates on my phone. But it was more out of necessity as I have a cr*p phone that is not able to use the SD card for storing apps for some reason, and the internal storage is pretty small, and of course there are many preloaded apps that can't be uninstalled, many of which I never used anyway (but because they are preloaded, taking an update would mean you're storing both the uninstallable preloaded version as well as the update). Basically if I don't manage the app updates myself my phone would run out of storage long ago, on app updates for many apps that I never use but can't uninstall either. I do manually update a few apps that I do use all the time.
Maybe it's different for iOS, but many Android phones, especially cheap ones outside of those made by Samsung and Google, tend to rarely ever get OS updates if at all. It's just app updates. OS updates tend to require the phone to reboot afterwards, so are you sure it's not just app updates?
As for the battery--how sure are you that the battery you purchased from eBay is in fact "new"? "Weak charging detected" does sound like the battery is old. I'm unfamiliar with the technical details of batteries, but it's quite possible that when the phone reports battery is charged 100%, it just means it detected it's not taking on more charges anymore, it doesn't necessarily mean the 100% is same amount of charge as the 100% 2-3 years ago when battery was still new. So an old battery wouldn't mean you can't get it to 100%, but that the 100% you get is less actual amount of charge than on a new battery, so it'd run down to 0% faster.
When you said it's losing charge "even when I'm not using it", do you really mean the phone is off-off? A quick press of the power button just turns off the screen, the phone will still consume power from the Wi-Fi radio, the cellular radio, plus any background tasks your apps and system may be running (such as, yep, those app auto-updates).
Depend on your usage pattern, perhaps it's feasible to leave Wi-Fi off most of the time except when you need it. This could even get side benefits such as the apps no longer auto-updating (generally you have to opt into a setting to let app auto-updates occur over cellular; make sure it's off lest you waste cellular data over those unwanted app updates).