Have also had such strange issues with worn out batteries on my Thinkpads. If it's consistently fine on AC, could be that the weak battery is struggling to output enough power for everything to spin up at once.

I've purchased two Kingsener batteries and both have the same or greater capacity as the originals. They feel much more substantial than the cheaper no-name batteries. Only issues are that the one in my X230 is slightly loose and it doesn't play well with charge thresholds after installing coreboot, i.e. it falls out of calibration and randomly shuts off around 30% if I don't let it reach 100% every 2 weeks or so.

if having to plug it in before turning your laptop on is fine with you, then you could probably just leave it alone until the battery completely dies.

i say this because -- back in the 1990's -- changing the battery sometimes caused problems like this and you might be better off letting this sleeping dog lie given the age of your laptop.