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See Kelsey Houston-Edwards's exceptional breakdown of Williams' paper, & Scott Aranson's thoughts on the topic.

[1] https://www.youtube.com/watch?v=8JuWdXrCmWg

I think the summary at the beginning of your first video is misleading; it's not a way to "trade space for time", at least not in an arbitrary program. The real statement is a bit odder to wrap one's head around -- "every problem solvable in t time on a multitape Turing machine is also solvable in close to √t space".

For a Turing machine that already solves a problem in n time and √n space (in other words, a lot of them!), it doesn't say anything.

When you convert a generic Turing machine into a Tree Evaluation instance, you end up with square-root space with respect to the original runtime t, but the new runtime will be far, far slower. IME, with these types of circuit reductions, the runtime typically becomes exponential in the space required, which is just about 'as long as possible'.

If we're being pedantic, it's trading time for the space guarantee.

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