That's kinda unusual that multi I/O cards die. The HDD and floppy that old are more likely to. However, try to reseat the cards and cables on both ends and see if things improve.

I did try connecting and disconnecting, and it looks like it works, specially if the computer is sideways (card pushed by gravity into motherboard). I wonder if there is any cleaning product or something I could do to get better contact?

Sure, you could use a fine eraser to clean the cards edge connectors until they get a shiny look, then clean with IPA.

There's lots of ways a 486 can be glitchy but still fixable.
Dirty/loose connections, including any socketed DIP chips, bad solder joints, marginal power supplies, RAM, misconfigured or overly aggressive jumper settings or BIOS settings.
Then there's all the old components that may be on the edge of malfunctioning.
The good thing about having so many socketed chips is that they can be swapped out, if you have reason to suspect them being bad. But the bad thing is it creates lots of places for bad connections.

Some of the first troubleshooting steps I'd try on any 486 would include checking voltages with a multimeter, and try disabling L2 cache.
If you have an intermittent problem, try touching/tapping different chips on the board.
This is somewhat general advice - in your case it sounds like you may have already found the problem.

For cleaning edge connectors (on game cartridges), I used to use Brasso but it might be too aggressive, I think it can scrape the gold off if you aren't careful. I've also used silver polish, thinking it might be gentler but not sure if it is. An eraser might be better.

DeOxit is made for this. Kind of expensive unless you'll be using it repeatedly in the future on other things.
The liquid version has a brush under the cap, you brush it onto the connector and then work it in and out a few times.
Or there's a spray can version. The spray is probably more useful for getting into the female end of a connector - like DIP sockets that the cache chips plug into.

I was struggling with intermittents on a 486 board a while back and sprayed Deoxit into every slot and socket on the board. It was a little ridiculous, the thing was dripping afterward. Later I reflowed some solder joints.
I think the problems on that board were bad connections in the cache sockets, and a bad solder joint under the keyboard controller socket, and maybe also one at the keyboard DIN connector.