My MacBook is turning off randomly, no matter plugged / unplugged

Hello,

my MacBook Pro late 2013 is turning off completely randomly, no matter how much charged is the battery, no matter if plugged in or running from battery. Sometimes it takes 2 hrs running, sometimes it runs whole day, sometimes 10 minutes. Usually turns off when I do something more power demanding, but sometimes not. I didn't find really any pattern to this behavior, seems completely random.

I have tried to reset SMC, PRAM, reinstall completely software (installed Catalina instead of Big Sur – in case if the older computer has problem with the newer system (as before Catalina worked perfectly fine and I had the feeling that Big Sur came with some smaller problems such as lowering battery capacity). I tried also to measure CPU and GPU temperature and this seemed also fine (when too hot, Mac turned on fan and cooled the computer down). For last few months, I have under the battery icon a message saying: "service recommended". I wonder if it could be a problem with battery not supplying the hardware with power properly? I tried coconut battery app to find out the condition of the battery:

I see from this that the battery is 19% charge capacity less from its full potential, a lot of cycle counts, other than that I don't know what to read from this. Strange thing to me is that it charges with 23.23 Watts only when a new original cord with 80 Watts is plugged.

The behavior is like this: sometimes the computer freezes at one point, I can't do anything, then the fans go on and then it dies. Sometimes it just turns black suddenly. After switching on again, everything that was opened reloads (it doesn't even ask for password).

I opened the lid, and it seems clean on the first glance. ( I removed dust, checked whether the battery is not swelling, there was nothing like traces of smoke, melting... and so that seems fine – but I am not well experienced and didn't dive deeper than that)

Do you guys have any idea?

The computer was repaired two years ago after a tea had been spilled into it.

There's a picture from CoconutBattery with a new battery:

I downloaded TG Pro as suggested and here's the result:

I got curious about this warning: "Last Shutdown/Sleep: Forced – Possible issue with the memory (RAM)". So I followed some guidelines for checking memory, using "Memtest 86+". No problems seemingly with RAM... This snapshot was taken just after testing the memory for like 3 hours, so therefore quite hot still.