In the Iwama system the "tilt" is a manifestation of the hanmi stance with all it entails. In the start of the second video, for example, it is there on the finishing cut preventing the ai-uchi (mutual kill) situation.
This makes sense. I noted the part in one of the Lewis videos where he talks about hanmi vs keeping both hips forward.
In Yagyu Shinkage ryu we call it holding your sword "naturally" or jun.
And if you've got your hips turned a bit, it is in fact natural to have your sword turned a bit.
Training to use the bokken as a bokken is actually really fascinating, in that a couple of hundred years of swordsmen engaging in rather serious duels with bokken to demonstrate their skills or the virtues of the system they trained, nobody ever systematized "bokutojutsu."
(Well, somewhere online is an essay written by Karl Friday speculating that such a thing must have been a thing, but there isn't any evidence of it).