Exactly for those reasons.

Most people can't pick out intervals by ear, but they can manage "higher" "lower" or "the same".

Think of it like a hash; it's actually important to discard specifics in order to be robust to changes in key & tempo.

Someone without rhythm and who's tone deaf could theoretically hum a song parsed through this notation and find the song they're thinking of.

Sure there's a possibility to identify a completely different song, but the more notes given, the smaller the chance to coincidentally find two different songs with the exact same ups/downs/repeats.

Time signature and key are totally irrelevant to the task of identifcation. Interval could be important, but vastly complicates the matter - identifying intervals is hard enough for many competent musicians.