The number of your attempts increase in fixed term, then the success rate clearly increase in broad perspective. It is equal to luck boost. This is the same as saving time like you said. You can try another map for time you saved and get better result in fixed time. Infinite time never exists, so I said "HUD affects records".
If you change "infinite" with "enough", it will better for comparing. If you have enough time for recording, your non-HUD demo will be pretty much the exact same as the one where you did use HUD. If you factorize amount of time spent on demo, then there is a direct affection depending on ones perspective. Not my perspective though.
I do actually not think using Kills/Items/Secrets counter is very unfair advantages in a single map run, excluding a very long map or a tricky map. However, HUD displays not only Kills/Items/Secrets counter but also Timer. I think Timer more effects a record in a single map run. You can compare your time with target time in some checkpoints while you are recording a demo, then you can find whether the attempt would beat the record or not in the early stage. The difference between a player with HUD and without HUD is obvious if the run is enough polished.
I don't get why timer is considered such a root of all evil. I believe even SDA speedrunners have allowance to be aware of their time, so why should it be c-n exclusive prohibition? Should I use my kitchen timer to ding after every 2 minutes? Would that be TAS? Kitchen timer is a devilish tool, after all.
If your memory is good, is it equal to doing a movie with HUD? No, it's wrong. For example, there is a route by which you can finish a map quickly but miss some enemies easily and the map has some viles. A player without HUD might select another route for safety because he avoid missing some enemies or going to a next map without noticing it for over 100% Kills. He also might spent time for checking the whole map again before the exit. In short, reliable routes for a movie would be very different between a player with HUD and without HUD because the success rate is extremely different. This example is not rare case. I think over 100% Kills by a vile is pretty troublesome. Your memory can't supplement the disadvantage in no HUD and a player with HUD never has risk of the exit with missing enemies. I think the difference is too large and too unfair.
You've just summed up why there was almost no interest in max movies as a category in c-n. Maps like to put player through annoying bits, it could be stuff from malfunctioning teleporters, unregistering secrets to revival of archviles. That sort of stuff was impossible to control without HUD, so it simply wasn't played and movies like that remained for the most part entirely ignored, and even if they were played, they were almost always overly cautious. HUD actually makes max movies worth undertaking, and I find no honor in forcing oneself through that oldschool annoyance ordeal.
By the way, I'm in no way trying to assimilate or force equal comparision between demos that were recorded with and without HUD. That's nowhere near my point. What I'm trying to say is something that frequent contributors to this thread most likely wouldn't exactly like to hear. Use your imagination what it could be, I don't want to come out as arrogant.