This is how the capacitors look, I was thinking that it may need a recap?
I dont know if they are supposed to be fat like that.

Possibly, they don't look too well - but I'm not sure if that's the only problem.

All I know is that electrolytic caps with 16v rating used to be very bad.
From experience, they failed very often. Maybe because they are usually forced to operate at 12v, near their maximum values.

Additionally, in a cramped chassis, under heat, they age faster than other caps, also.
Manufacturers often prefer to use them because they are both small and cheap.

Anyway, I have little to prove that. So don't take my words as face value.
It's more like a gut feeling, than an opinion/fact.
In the late 90s, there also was a capacitor plague (metal can caps mainly)..
However, the Sega Game Gear, which also often has defective 16v caps, was made years before.

If you can, clean the PCB softly with isopropyl alcohol and cotton sticks ("qtips"), as well.
This makes soldering a bit easier, also.

Oh, and please don't despair. Fixing such things may take a while.
Also try playing an Audio CD, if you can.
This needs no computer, also. So you don't endanger one of your vintage PCs.
Just make sure the AT PSU (if you use one) was enough load on the 5v line..
An old HDD will cause enough power draw.

At least the disc spins up. I don't know if it's true of all drives, but the Panasonic 562b will only spin up after it's got a focus lock. Then once it's spinning it tries to get a track lock. As well as checking the capacitors (agree that they don't look great), might be worth making sure the read head hasn't jammed and it moves smoothly on its rails.

Those caps are leaking, and I see some corroded vias. From this angle, it doesn't look like it would be that hard to fix, though. Those are low-profile caps (5mm or 7mm tall), so make sure you get caps that will clear the mechanism above them.