> the idea that there are the same number of integers as even integers is a stupid one that in the end does not lead anywhere useful

it leads to the idea that measuring 2 sets via a bijection is a better idea than measuring via containment

That a bijection exists is incredibly useful. But the idea of "measuring" infinite sets in the cardinality sense is not very interesting or useful.

Saying that two sets have the same cardinality is equivalent to saying there is a bijection between them. I don't understand how the latter can be useful but not the former?

It isn't very interesting or useful... to you.