I figured Game Genie would do that, but I feel like the hacks would be made for people who don't have Game Genie or games that don't support infinite continue features.
These type of hacks are made for old games (like starting at the NES era) that only give you a limited amount of continues. The reason why I think this idea is unpopular is because I feel like people would complain that it makes games too easy. I think infinite continues make games more accessible.
It is not an unpopular, as in looked down upon, concept in the gaming world at large.
If it is unpopular as in volume of such hacks compared to other hacks a game might have, or relative to hacks in general, it is likely more because most go for infinite lives or assume people have savestates and thus don't need it.
Certainly there will be elitists out there but few care about them. I would mostly only expect to see such a thing made for those that want to play the game but also if using a continue resets your score/acquired powerups or something such that someone can play for score while others play for fun.
Some have speculated why a game might not have infinite continues. Some pondered whether it is a holdover from the arcade (which would have been the dev platform/where all the big boys played at the time). Some have pondered whether it was to give games staying power. I will leave that one be this time though.
For the record as well.
Game genie (at least as far as 8-16 bit era things are concerned)
A device which rewrites part of the ROM (usually by intercepting reads to it and sending its own back).
A game genie code can then be trivially patched into a ROM as said game genie code is literally the address of the part you care about and what needs to be there.
The game genies tended to have limited abilities to write RAM.
Action replay (also code breaker, gameshark and a few other things, if you speak to Chinese people they might call them goldfinger).
These primarily write to RAM, which tends to be where ammo/mana counts, lives, score, time, item counts,.... are stored.
They might have the option to alter one or two short things in the ROM, though these were mostly intended to bypass protections.
This is not trivial to automatically patch into the ROM on older systems. Newer ones with a bit more overhead (the GBA and DS are good examples) can have an additional thing automatically added to mimic the classic RAM writing device thing. A vaguely competent assembly hacker should have no problem porting a RAM cheat over; you either find the instruction/routine that uses this area and nerf it, or find a way to add a routine such that you basically do the RAM cheat thing again and it constantly writes a value to that area.
The names got a bit blurry as time went on and companies were bought and sold -- sometimes said companies would come back with save dumpers and editors, sometimes they would do region free, on the likes of the DS then the binary was loaded into RAM and thus a RAM cheat could alter things at a fundamental level... classically though and for most people around here the above distinction holds.
Other than that I already covered how to make a cheat to fiddle with the continues, and some shortcuts to make it a bit easier and some things that dodge the problems with it (if you only have three continues you can't do the braindead easy use one, search for lower, use one, search for lower... thing and expect to end up with a manageable list of results to go through).