Suppose I had a prior for the probability of some event C of, say, 0.469. Could one object to that, on the grounds that I have assigned a probability of zero to the probability of C being some other value?

Depends on how you're doing this; if you have a continuous prior for the probability of C, with an expected value of 0.469, then no— and future evidence will continue to modify your probability distribution. If your prior for the probability of C consists of a delta mass at 0.469, then yes, your model perhaps should be criticized, as one might criticize Rosenkrantz for continuing to assume his coin is fair after 30 consecutive heads.

A Bayesian reasoner actually would have a hierarchy of uncertainty about every aspect of ver model, but the simplicity weighting would give them all low probabilities unless they started correctly predicting some strong pattern.

A prior of independence of A and B seems to me of a like nature to an assignment of a probability to C.

Independence has a specific meaning in probability theory, and it's a very delicate state of affairs. Many statisticians (and others) get themselves in trouble by assuming independence (because it's easier to calculate) for variables that are actually correlated.

And depending on your reference class (things with human DNA? animals? macroscopic objects?), having 2 eyes is extremely well correlated with having 2 legs.