Emulators have been a divisive topic for quite a while. The majority of games on consoles that have emulators available allow them, but some games do not, for various reasons.
One common reason for banning the use of emulators is that the emulation is not always accurate on certain consoles. N64 emulation has been quite bad for a long time, with certain games not running at all in some cases. Genesis also does not have great emulation in comparison to consoles like NES and SNES. Some games ban the use of specific emulators, for reasons such as inaccurate framerate in comparison to actual console, or not handling RNG in the same way as console.
It is also quite trivial to cheat on emulator in certain games. The inputs can be scripted to allow for perfect execution, and some emulators like FCEUX for NES even have tools built into the emulator to allow automated inputs on each frame. Memory addresses can also be edited to allow for ideal RNG, making parts of the game faster. This is a bigger issue for people verifying runs than spliced videos, because it is often extremely difficult or impossible to tell if someone has used these tools instead of playing the game themselves.
There are many arguments for allowing the use of emulators though. One of the arguments that is brought up a lot is that letting people use emulators makes a lot of games more accessible, as they do not need to buy the game, a console and a capture card setup to record their runs. For some games this is an absolute necessity, as the rarity of the game makes it too hard to get your hands on a copy. This is not always the case, however - the expansion kit for F-Zero X costs over $1000, for example, yet the use of emulators is still banned. This is going to become a huge problem as time goes on, as games and consoles will eventually stop functioning properly and become rarer/more expensive.
Sometimes the argument extends beyond reason. Recently there has been a discussion over the use of emulators in Metroid on NES. It seems that some runners were using save states on emulator to reset the game to the title screen at the beginning of each run. The issue with this is that the game loads in all of the random elements like enemy spawns, item drops and patterns at the title screen, so every playthrough of the game is exactly the same without any difference. It means that you can keep resetting the game until you get a playthrough with perfect RNG loaded in at the title screen, and then just play that save state each time.
Rather than ban emulators, you could just get people to use an overlay to show that the emulator was reset and did not load a save state at the beginning of the run. I was watching some Metroid runners discuss this a while ago, and something bizarre happened. They started to test manipulating the RNG in Metroid by changing the console that was being used to play it, and seeing if the amount of time waited until reset and the chips in the console made any difference on getting consistent enemy patterns in the game. It did. So if it is possible to manipulate the RNG in Metroid just by having an NES with the right chips in the console, should those consoles be banned too?
It is a very tricky subject to discuss, as all games are different. I do think that the vast majority of games should allow emulation though, it's just going to become a necessity over time anyway when it is not possible to play them on console.
Tagging emulator runs already exists, I think it is on every run submission form on SRC. Rather than ban emulators completely, some leaderboards split them up into different categories, I think SM64 does that. There are reasons for splitting up consoles/emulators in some cases, like OOT glitches being possible on virtual console but crashing the N64 when performed. Goldeneye is a good example of a popular game that does not allow emulators. This is because Goldeneye speedrunning has a long history of cheaters, N64 emulation was not reliable at the time people first started sharing runs and the community is kind of a separate entity from other speedrunning communities. They have a page on SRC but the main record keeping for IL runs is not on there.
The whole concept of standardisation started with Twin Galaxies and Speed Demos Archive. Neither TG nor SDA allowed glitches, for example, which obviously seems ludicrous today. Some games still use something referred to as "SDA timing", which means that the timer should start on the first frame of player control.
I think certain forms of standardisation are important. NES speedrunners tend to be variety runners, due to the amount of short games there are. Banning emulators as a standard would mean far far less runs being added to leaderboards, but things like standard timing methods, no cheat codes, no turbo controllers etc is a good thing.
The argument has been simplified to "accessibility vs authenticity". You will find that those in favour of "authenticity" are people who already have good recording setups for consoles and a ton of games, or only play a couple of games. I think in general, emulators should be allowed and standardised rules should be applied in cases where they make sense. If there is anything that does not make sense, or there's an issue that needs to be addressed, let the individual communities sort it out. The generalised argument some people are debating makes no sense to me, it's stupid to try and apply rules to games you know nothing about when there are people who've played them thousands of times and know exactly what the ruleset should be.
This is kind of what it boils down to, but in this case the people who set the rules are the people who play the games, due to the way the mod hierarchy works.
Edit: the monitor/retroarch debate would be an interesting one. I think it wouldn't really make a lot of difference though, because the majority of speedruns are pretty much the same each time, there is no reaction, just muscle memory. If I started playing SMB on a setup with a different amount of input lag, it would be a hindrance at this point.