You don't get the luxury of hand waiving: the two are equivalent, and thus any rigorous proof in one has an equivalent proof in the other. In acknowledging this, you accepted your original claim was false.

So what you really meant here was:

"Ofc, geometrical proofs are technically algebraic proofs (due to Homotopy type theory). I must have had a brain fart when I implied that one could be more, or less real than the other. That made no sense."