Hey, I’m curious to see whether you have the same issue as most people here. When the phone shuts down, does it display low battery logo, prompting to plug in to charge? If you plug it in and turn back on, does it show the (roughly) battery percentage at which it switched off? If you then enable the Low Power mode, can you run the battery all the way down to 1%? If you use Low Power mode all the way do shutdowns still happen?
Battery replacement costs $80 or equivalent in local currency but I am not sure the issue discussed here can be fixed with battery replacement.
I just asked my wife and she said that it does show the low battery logo and promting you to charge the phone, after trying to turn it back on, when it shut off. It also was at the percentage or right around at which it turned off when she plugs it in. She hasn't really used low power mode at all, so I can't answer that question. I'll have to tell her to try that next time it happens.