EDIT: I'm retracting my suggestion. I'm leaving the post up for posterity, but after watching the top NG run I've seen why the runner choose a different method for completing the opening of the run.
I'm very new to the speedrun community, and decided I would try my hand at No More Heroes. While waiting to get a working wiimote, I started researching a few existing speed runs. I was comparing the current place holder's speedrun to the summer games done quick speed run. The current record holder has a time that is almost an hour faster than the summer games speed run, BUT, and this is a big but, the summer games run goofs off and also allows several cutscenes to play for backer support.
I have noticed that in New Game+, the summer games person makes ample use of charge attack and more importantly the jumping attack travis can do (I'm not sure what its called or the input, i just know its a lokilov ball move). Just comparing the jump attack method to the current world record holders method of using wrestling moves on the first two enemies in the game, the jump attack finished in 3 seconds whereas I think the other method needs around 10 seconds.
If this method can save that much time, and is used on alot of enemies in the game, I think it is possible to cut down this speed run by alot. I don't know if I'm going to be the one to do it, as it will be awhile before I have anywhere near the skill of either of these speedrunners, but I'm going to try to get there.
For those interested in beating the current time for New game+, use jump attacks and charge attacks (save charge attacks for enemies that take time to spawn or for groups of small enemies). Later in the game the jump attack is not an insta kill, but after one or two hits it can finish off any enemy while skipping the coup de grace and roullette.
It's probably not obvious what I'm doing in my NG+ PB because I don't think I reference the purpose of the wrestling moves at the beginning. Wrestling moves manipulate the slot machine in some way, although we're not sure of the specifics. I'm not actually sure if it's worth it in NG+ Sweet since it only saves a small amount of time if you get it instantly; in that PB, I was a little frustrated about resetting for a 777, so I just kept going in spite of the time loss.
Let me know if you have any other questions about NMH1 speed running. I'm always willing to help out new runners.