I own a PSP 3000. I'm sure the battery is pretty much useless now. I keep it around for decoration but I haven't played it in ages. I went full digital as much as possible (a few of my games were UMD only but every physical game I owned, I bought again digitally). To be honest, I don't think it changed the battery life that much. It still ran about three hours on a full charge.
You asked about the analog stick. It works well enough but it did feel kinda weird. No other console or portable that I can compare it to. It just sorta slides instead of tilting.








