The 2019 change, clarified
Rabies is still required — the health certificate isn't.
On December 16, 2019, Mexico dropped the comprehensive zoosanitary health certificate — so you no longer hand over a vet-signed import certificate. It did not drop the rabies requirement: you still need a current rabies vaccination certificate and a parasite-treatment record, which SENASICA reviews at the inspection.
Flying instead of driving? Airlines often add their own rules — a vet health certificate issued within ~10 days of departure is common. Check with the carrier.