Does zero-padding the secret in Shamir's sharing scheme increase security?
As you note, Shamir's threshold secret sharing is perfectly secure (or information theoretically secure), yet does leak some information about the size of the secret (same thing with one-time pad).

If you are worried about leaking some information about the size of the secret, then padding could be used to lower the information leakage (instead of knowing that say the secret is 32 bits, the attacker would now know that the secret is no larger than say 1024 bits, something along those lines). If leaking the size of the secret is a problem, then this indeed increases the security of the scheme.

Padding with random bits vs zero-padding would make no difference in security. The secret is already perfectly secure (except for the size of it). Zero-padding hides the length just as well as random padding.

So, what if the secret starts with a zero bit? One thing to do (and this works with random padding too) would be to prepend the length of the padding to the secret. Other techniques would work too.