[finnw]

Breaking SHA2 (i.e. developing an economically-feasible preimage attack) would indeed crash bitcoin, but the converse is not true.

A near-collision attack on double SHA256 (if you treat it as a single hash not a pair of independent hashes) would also crash bitcoin, but would not necessarily be a threat to use of SHA256 for authentication purposes.

A bitcoin block solution just needs the hash to include enough leading zeros. Authentication (nearly always being automated) requires every bit to match - hitting 255 of 256 bits is no better than hitting 0 bits, as either way your message will be rejected.