Why is counter mode encryption with a 16-byte cipher block not broken?
You are right that it is generally foolish to try to use AES-GCM for zetabytes of data under a single key.  The standard ought to have clearer data volume limits.

For a long time, the folklore understanding was that a non-collision in a CTR stream doesn't really leak all that much…until sweet32 demonstrated that it actually can be a real problem.  Still, what is feasible for a 64-bit cipher with online cost ${\sim}2^{32}$ may be considerably harder for a 128-bit cipher with online cost ${\sim}2^{64}$.

An alternative is to use crypto_secretbox_xsalsa20poly1305, which is a pseudorandom function family rather than a pseudorandom permutation family like AES, so the generic birthday distinguisher for permutation-based CTR doesn't apply.  One could also imagine using a wider-block cipher like Threefish-256, in which case the birthday bound is unimaginably larger; it is a pity that AES was standardized with a meager 128-bit block.