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.
I think a lot of the issues you're bringing up are already addressed by the way categories are divided. If you don't want to run a category that has glitches, there will probably be a glitchless or no major glitches category. In the event that the category you are interested in doesn't exist, you can run it and submit it with a request for it to be added to a new leaderboard. Forums exist for each game to discuss what kind of rulesets these categories might have, as do resource pages and guides that can have things added to them that may not be apparent to someone watching a run of the game for the first time.
There is also some overlap between speedrunning and things not quite related. The quest to complete SM64 in as few A presses as possible is an example of this - not strictly speedrunning related, but a strangely popular topic in speedrunning circles. These things just crop up in places where people who are interested in running a game can find out about it, it doesn't necessarily need to be on a leaderboard.
Some games also keep their own leaderboards. The most high profile one I can think of at the moment is Mega Man. I'm not sure what happened, but the general gist of it is they wanted some features that aren't present on SRC or something.
I think the best way to handle the leaderboards is to make them simple and easy to use, anything else is going to put people off running games. Having an agreement to run the games in certain ways helps with this, and also makes the leaderboards more competitive. How do you even figure out what the rankings are or what time is a good time if nobody is playing by the same ruleset? The actual content of runs is going to end up so removed from what runs in a centralised leaderboard are like that there is no point in having a leaderboard at all.