Are hybridized orbitals used to explain the fact that the four bonds in methane are equivalent? If not, what is?

Answer 1

Yes. C is 1s2 2s2 2p2.
One of the 2s electrons is promoted to the 2p to make C 1s2 2s1 3p3, then the s and p orbitals are mixed to make 4 sp3 hybrid bonds (about 1/4 s character and 3/4 p character). All of those bonds are equivalent.