If your thought experiment involving pieces of paper has helped you to think of useful or interesting things that you wouldn't otherwise have thought of, that's great. It hasn't so far done anything of the kind for me. The fault could of course be mine.
What I said was not (as you claimed in another comment in this thread) that I don't see how constructivism is relevant -- though in fact I don't think constructivism is relevant -- but that I don't see how your thought experiment about pieces of paper is relevant.
I regret that it wasn't immediately apparent to me that your thought experiment was about constructivism. Now that you say it is, I can kinda see how it kinda relates, but for me thinking about your pieces of paper doesn't conjure up any useful insights about constructivism, and if I didn't already know about constructivism I don't think it would particularly point me in that direction. Again, if it works that way for you, good for you, but I think that when you present it to other people you are forgetting that the others (who unlike you haven't been thinking about this picture for 30 years) don't have the same associations with your imaginary pieces of paper that you have.
I have no problem with the idea that one might want to construct alternative number-like objects. (And yes, I know what Cauchy sequences are and what one does with them. And no, they really don't have very much to do with the parallel postulate. There's an analogy between the parallel postulate and, say, the LEM, but that's all.) But (1) again, this has nothing much to do with Eric Lengyel's critique of how geometric algebra is presented -- when Lengyel talks about "foundations" he isn't talking about going down to the level of set theory, logic, HOTT, or whatever, he's talking about getting the basics of geometric algebra right when one already has notions like "real numbers", "vector space", etc., in hand -- and (2) your thought experiment happens not to do anything for me to clarify, inspire, etc., ideas about alternative number-like objects. (Also, it seems to me that if you want to use that thought experiment to say something about number lines then you want your 0-width lines, half-planes, etc., to lie across the number line, not along it.)
It is simply not true that book 2 of Euclid "defines geometric algebra for the first time" or indeed defines it at all. Euclid is doing geometry but is not doing geometric algebra, which is something more specific.