Regular MACs vs Carter-Wegman MAC
A lot has changed recently in this area. Now the only ciphersuites Chrome considers non-obsolete (those that use AES-GCM or ChaCha+Poly1305), do use Carter-Wegman MACs.

So, I would say that there is no disadvantage and that any low popularity has been just an artifact of historical decisions in standardization. Secure hashes were the first to be openly standardized and exportable, so they were used for MACs as well.