The proof then remains a valid proof of something although not exactly a proof of everything its authors claim it to be a proof of.
An erroneous proof is also usually useful because it is not clearly wrong in most cases since nobody has come up with counter-examples so far that would prove the proof incorrect, yet.
I think it's more that the mathematician has a top-down way of reasoning, where they can see things like "I want to get from New York City to Los Angeles, so I have to board the bus, take a flight, and then take the bus from the airport at LA". There are certain parts where you basically know that a proof will be possible, because it seems true, like "I can get to LA's airport with public transit", so usually a specific hiccup, like a bus being delayed, won't prevent you from getting there.