I'm not speaking (only) of non-constructive proofs, I'm just saying that sometimes a proof will leave you with no real sense of why a result is true (or "inevitable"). That sense sometimes comes much later, after further results are known.
There's a notorious proof of Heron's formula (computing the area of a triangle from the lengths of the three sides without trig) that proceeds by proving three things that seem to have nothing to with one another or with the goal, and then through a flourish of algebra the result falls out. There are much more straightforward ways to prove the theorem that do not abuse the reader, and modern mathematics frowns on such indirectness unless it is totally necessary. Unfortunately it was the style to write gimmicky proofs like that for a while in IIRC the 18th and 19th centuries, and sometimes they persist.
