Here are some news !
I received fresh capacitors for the HDD and japanese caps for the PSU. I said that the computer crashed randomly, well now it doesn't do it anymore. It's funny to see that 2016 rubycon capacitors are actually 2 times smaller than 28 years old teapo capacitors. However the functionning of the hdd is ... random ... most of the times it seeks and stop. But sometimes it manages to not stop and then it lets the computer boot on it and the computer then displays a lovely
C:\
>
(which I can't use since I don't have any compatible keyboard. I tried the arduino converter, but I had no luck with that thing ...)
Oh and also when I said that this PC was in an AT form factor, well I was wrong. I just didn't knew how XT looked like and since it looks a lot like AT I thought that it was AT.
I'm leaving the computer turned on doing nothing because I think that it may help the hdd to work a little better
Edit : I left it for approximatively 4-5hours almost countinuously. After one hour, it still did the same thing. After two hours, still the same thing. I was a bit sad about that thinking that it would stay like that ... But I kept it running during the evening and when I came back and rebooted it once again, it worked on the first try ! Second try : still working ! After a 15 and a 30min break, still working. I'll see tomorrow if it really repaired itself or not
Edit 2 : Okay so it was still working this morning. At some point it started to do the same problem, but after running it for half an hour it came back to life again. I tried to boot from a floppy disk in order to control the computer via hyper terminal as someone advised me, but it doesn't want to for some reasons ... maybe because this controller is also the RTC card and so it was in a bad shape after the leaking of the awful varta battery. I didn't notice last time that the jumper box of that card has some switches that move just like there was nothing behind it and some other doesn't want to move at all so I think that this jumper box is completely destroyed in the inside. I will replace that. I'm also going to replace some of the chips on it because some of them has remaining corrosion that may (or will) cause issues too