"Well, continuing the C chord thru the A note of "birth" doesn't exactly sound awful, but it sounds lazy to me. Why not switch to an F chord for the duration of that note?"

Alternatively you could see the shift to F as clumsy and unnecesssary.

The A is the 6th of the tonic chord and as such it would create a Cmaj6 chord which is a static chord and needs no justification in the key of C so the A wouldn't even qualify as a passing note.

You could perhaps use F/C (F in 2nd inversion) thus maintaining the C bass and using the F as a passing 2nd inversion IV chord (known in classical terminology as a passing 6/4 chord) and in all probability that was the original intent of the composer, however the Cmaj6 produced by continuing the C chord serves the same purpose in this context from a more advanced perspective.

innit ...