But there are advantages to controllers too -- it tends to be easier to press two buttons on the same frame for eg.
And the thing is, it's not like there's only one controller you can perfectly standardize on. Even if you require only an 'original controller from the console manufacturer', well does that mean you have to be using the original rubber contact pads? Those are mostly all dead by now. Are all replacement rubbers equal? What about putting tape in the middle of the dpad to make diagonals more consistent?
And then it's also an accessibility issue. A lot of people can't use the original controllers. A lot of people who are otherwise very good at video games. Many of them can't use them because they played so many video games on them and they've gotten RSIs.
All that said there are usually limits. Almost no games allow macros (exception: Celeste allows a macro for a dash technique) or any other "press one button for multiple inputs" kinda stuff. Very few games allow turbo (exceptions: some RPGs do because mashing through cutscenes is very bad for your hands).
A lot of games do require controllers that are at least no more capable than one that came with the console though. But then there are things like one-handed controllers that are no more capable but could potentially change what's easy/hard.
So.. yeah it's complicated. It's just down to the game and its 'community' to decide what makes sense. That's largely how the whole speedrunning world works these days.
But no it's not too late to change the rules. Speedrun leaderboard rules change all the time to accommodate new information, techniques, and glitches. If people who run SMB feel it's unfair, it might change.
An example of this would be if the original controller won't let you press both left and right at the same time, and being able to do so gives an unfair advantage due to the way the game was programmed (I think Castlevania SOTN is one of these.)
Though if the controller pcb is really worn down by the rocker's center nub, it can start to become possible, and then there really is nothing stopping you. It's not uncommon for really well-loved controllers to end up like this.
It's definitely a letter-of-the-law thing where you could claim "but it's an original controller!" but no speedrun leaderboard is gonna let you get away with it.
Also I don't think emulators do a neutral input if you press both, that would be quite weird (and unlike how it happens with a real controller when you move from one to the other). Instead I think they prioritize the first input.
The way emulators handle L+R depends a bit.. For SMB1 specifically you're allowed to use an up to date version of Nestopia, Bizhawk, FCEUX or OpenEMU, all of which by default give neutral inputs if you press L+R (I don't recall if OpenEMU works this way but I believe it does.) There are other emulators like older versions of Bizhawk which gave priority to your latest input, or older versions of Mesen which gave priority to left and down. I'm sure other methods of handling L+R exist as well.
Which method is most correct is debateable, and sometimes contentious! :)
For some games there are tons of categories for different variations or restrictions. (It can even happen that when a new fundamental glitch is found, a category "Game X without this one specific glitch" is added if people enjoy running the game without it anyways)
If you're using an emulator, you have to enable allowing it. If you're using a hardware adapter you'd have to have your adapter blocking it.
Though an original NES controller speedrun coming close should definitely be a legit accomplishment.