Finding alternative proofs is not a valid way to confirm the original proof; so... no? You appear to be absolutely correct that no one has taught you what math

Finding alternative proofs is not a valid way to confirm the original proof; so... no? You appear to be absolutely correct that no one has taught you what mathematics is. There's definitely a failing here in mathematics education; unfortunately this is not a surprise at all. Validating the proof requires understanding what it is you're trying to prove, understanding what are valid proof techniques, and understanding the premises. You then check every step against this knowledge. It isn't easy, and tbh I usually couldn't be bothered after spending hours on a problem, but this is where results are obtained in proportion to effort.