After a couple months of development, one submitted master's thesis and one burnout, I'm pleased to say that I now have a formal proof of correctness of
@dwarnDavid Wärn's zigzag construction of identity paths of pushouts, using the type-theoretical definition from his first note.
https://github.com/UniMath/agda-unimath/pull/1370
AFAIK this is the first completed formalized proof that it satisfies the universal property of identity systems, in any proof assistant. I'm pretty proud of myself, ngl.