99% being ignored is probably generous. Percentage is higher still.
Every hacker is inherently short on time and long on ideas of their own, playing a new game usually only adds to that list. Doing someone else's does not make their list get any shorter. Any projects that do get taken on are ones that personally interest the hacker, ones that might take 5 minutes, and sometimes things have already been done so people can point to them. Personally I visit this thread for game suggestions mostly, and will occasionally dash out a general overview of what such a hack would entail and an approach (or indeed why it is not likely to happen either at all or without Herculean effort, bordering on Sisyphean). You probably also have to figure how many people are playing on each console around here -- PS1 and PC is not the most popular, but more so than some, and PC at least has some wonderful tools for debugging (PS1 is not the worst either).
This seems like a reasonable request, and as it is essentially doing things the game already does then might not be the worst thing in the world.
Doing it properly would probably be a hard thing for a first project.
Doing it with more of a cheat focus is a different matter, you might even be able to spin that up into an edit you do externally. This would mean randomising cheat values each time, or same for the edit but fairly reasonable thing for most things running ROMs.
Anyway I would probably start by doing a character selection cheat for you and player 2. A great number of fighting games will just have player 2 (and beyond) being AI and not something special/extra/different.
Character selection cheats are usually not that bad to make and are made at the selection screen, or just after. Bonus is this can also unlock hidden characters, beta characters, sometimes play as bosses (more on that shortly) and the like, and works on things with a character selection screen in general and not just fighting games.
Three possible approaches that might work or become necessary depending upon the game and way it is coded
1) if you are lucky then merely hovering over the character is what does the deed. Flick back and forth doing search for change, staying still and searching for things that stay the same.
2) As above but you have to press a or whatever to confirm your selection
3) As above but you have to actually start a match with all players confirmed. Do remember you have savestates so you can happily set other things up and only change the character selection.
Options for playing as bosses for reference tend to be either just after the last player character numbers, just before maybe if that is a thing or slammed right to the end of the range. That said you might have to try everything in the range.
Knowing both of these locations and the rough values they can have you can start on the tower. Check that player 2 indeed does change accordingly (remember you have cheats so give yourself full health/crazy damage/whatever cheats there are and savestates to go back if you mess up)
If you are lucky there is a simple line of data in the ROM that gets filled in for each successive battle and you can randomise this.
Personally I would bust out a debugger at this point and do a break on write for the second player character selection. You could try noting what each successive battle is though and searching for that.
If you find this data and it is indeed data* you can edit easily then great, hopefully you can do the cheat thing or ROM edit from here yourself. You could do a cheat to change the current player randomly (or in an order)
*any schoolboy programmer would recognise such a setup as you describe as candidate for linked list and while I might not bet on the 16 bit era efforts at mortal kombat having that then PS1 and PC was firmly in C programming country. If indeed it s an array like your quoted section then while rather different devices I would wager more on the array side of things than linked list family or other possibilities.
Anyway if you can't find the list by normal searching then you will have to play debugger. Whatever writes the selected opponent player will stem somewhere along the line from the ROM somewhere. Welcome to tracing. I don't have a nice guide for the PS1 but it is much the same whether it is a commodore 64 or PC game released yesterdayhttps://www.romhacking.net/documents/361/
You might get away without following it right to the source if you find a more whole selection of data to search for in the ROM.
If doing a proper hack you would instead change the game's code to pick a random one each time, or whole selection if the game works that way. Bonus is the thing you just did with characters is a direct result so should not be too hard to subvert.
I don't know what will happen for stages in this but probably something similar will get to be done, and what results in character selection will most likely trip the stage selection value at the same time. If you happen to find the stage value as part of this then you can probably also make that a cheat if you wanted.
For someone versed in the techniques above it might not even be that involved a project. If you are learning, and it would be a reasonable one to learn with, then maybe a bit longer.
Not sure what we are suggesting for PS1 debuggers these days (don't know if no$psx ever got something nice here, the other no$ debuggers for other systems are if not the gold standard debuggers then seriously up there). PC stuff varies as well -- if it is PS1 vintage then it might well work with older tools like ollydbg without having to do more fun things like radare2, ghidra or even busting out IDA (the free version includes older PC standards so there is that), also not sure what we are suggesting for basic cheat search on PC games (every time I wander into PC cheats world I find myself shaking my head that people pay not inconsiderable money for things debuggers and other such tools will do better and for free).