iPod Touch won't turn on

When I used to use my iPod Touch for music, the device had to be connected to some sort of external power supply in order to be able to use it, but randomly one day, when it was connected to an external battery, it shows the apple logo, then turns off. Since I was told by apple that there was no kind of repair process that iPod Touch can go through (unlike the iPhone), I just got a 128 GB iPhone 11 to replace this device.

I also have an iPod Touch 2nd Generation, but it never just stopped working like the the 6th Generation did, so what gives? I never dropped it, and all I'd like to be able to do is make a full backup that I can then extract to see what exactly is missing from my iTunes library. Thanks