Charged battery, but doesn't even try to start.

Sounds like car may have been running off battery and pulled it down. Especially with it wanting to die. Now the issue is did the charging system have an issue that killed the battery. If you have a volt ohm meter check battery voltage after charging and then see what voltage does when trying to crank the engine, I’m betting it drops off significantly. Could also have load check done on battery. Could be a diode failed in alternator which could be allowing battery to drain.