You're right--punching is not required. The glitch is caused by making the automatic jump wait. You jump just before Arnie jumps out of the machine by himself, causing him to just jump up and down. When he hits the ground, he starts jumping forward. Then, when he would've hit the ground had you not tampered with the intended programming, Arnold starts walking like as if he was on flat ground. Instead, he's still in the air. This looks cool and all, but every pixel or two upwards this glitch gives requires a frame or two of jumping beforehand, so the glitch requires sacrificing frames. I've got an .fcm where he's definitely walking on air (though only half as high as in your run) and it only costs 6 frames. Given the game in question, I think it's worth it to add. Good idea.
When testing this, I ended up with one .fcm where Arnold first ends up behind the smash effect, walks really high off the ground, and then never leaves the stage because, I guess, he's too high. So he just walks into the wall a little bit, and the whole game freezes. Jolly good!
another test: Invincible Arnold!
http://dehacked.2y.net/microstorage.php/info/1473033982/Total%20Recall%20%28U%29%20trest.fcm