Thanks to you all for the in-depth replies, I really appreciate them, I'll try to reply to each of you:
@rhb:
That would have been a fantastic solution when I was troubleshooting a SEGA System16B board (~1980) as the system has just a couple of CPUs (Main cpu Motorola 68000 and a Zilog Z80 for audio). In the end I "brute forced" by checking every single thing with the continuity function of the multimeter and discovered that two address lines were tied together. It was just a matter of following the trace then until I've found a scratch that joined the two adjacent traces. It wasn't an easy feat by any way, those things have shared buses which are the real fun police when looking for defects.
Unfortunately, starting in the early 90s, big companies like Capcom grew tired of seeing their own hardware bootlegged and started developing new boards with the real repair showstopper: huge custom ASICs.
The board I'm working on is made of two layers: the mainboard and the ROM board and I've managed to isolate the problems and those reside on the latter. The problem is that this board has 4 HUGE customs, surface mounted and with a gazillion pins each. By doing pure guesswork I've managed to discover that one manages the sprite layer (OBJ data), one the background layer (SCR data) and the other two I think that involve something with the audio and program data. The problem is on the sprite layer, which helped me further to isolate what the possible faulty components might be.
Those chips are, obviously, completely undocumented, so you can only guess what they are doing (probably doing addressing), which means that I have no way to properly test them, even a logic comparator would be useless.
What I'm doing is basically checking EPROMs, RAMs and other ICs (74LS245 etc...) to see if they are working properly, hoping the problem isn't in the custom ASIC otherwise that's game over.
A logic analyzer... I'll have to do some research here, have to say that this forum is truly a gold mine.

@FrankBus:
That's a very interesting project and your DIY logic analyzers are really, really impressive, it would be a while before I can make something like that.
Anyway, building my own would be surely a fantastic learning experience, but my free time is really scarce and I don't know how much time I could pour into this.
By the way, you're the second user suggesting me a logic analyzer, so I'll definitely look into them ASAP.
@KL27x:
I think the PDS1102 is an older version of the one I linked, which is the new SDS line and doesn't work with batteries. I'd definitely keep Hantek as a brand in mind, though.
I've seen the handheld Hantek 2c42 going for ~100€ which has 40Mhz bandwidth and might just be enough, with the nice plus of being small and battery operated (and it also doubles down as a multimeter, which is nice. Might become my jack of all trades).
@tggzzz:
Hopefully I won't sound arrogant here, but I've been repairing stuff for more than 10 years, ranging from Hi-Fi equipment to computers and videogame consoles and the *first* thing I check are electrolytics

They have fooled me more than once in the past: while I was repairing an old Technics SL-D1 turntable (1979) everything looked pristine, the caps were nice and shiny and still... they were dead. Electrolytics are definitely the first thing to check (and, if you're working on anything older than 20 years, replace, even if they check out good).
Again, thank you all for your inputs, they are really valuable to me. Now I've added a new variable into the equation: a logic analyzer, something I've not considered before because I thought that an oscilloscope would be a more "all round" tool to have.