Does a DCM flyback always operate in CCM during startup?
A lot of them do just that. The other option is to ramp up in DCM (with no dwell time which is obviously the border with CCM) very slowly and the reason is (and I suspect you know) is that charging the output capacitor up from 0 volts cannot be done quickly because the slope of the current passed into the capacitor is naturally very low hence, the energy transfer is quite small. 

The slope of the current (from the charged inductor) multiplied by its inductance equals the voltage on the output capacitor hence, because the voltage is close to 0 volts initially, the inductor cannot dispense much of its energy in the regular switching cycle.

If you tried to do it in CCM you could run up quicker but you might easily over stress the inductor.