Why can't the equivalent capacitance be calculated like the method 2?
The charge leaving the 10 volt source is basically the number of electrons that flow into the capacitors in order to charge them up. That movement of electrons is the same for all the capacitors and the source. Electrons are electrons and don't disappear.

So, the charge on each capacitor is the same as the total charge of the whole circuit. Charge is just the number of electrons and, the same number of electrons flow through all the series connected capacitors. If we were talking about three series connected resistors taking 1 amp, would you say that the total current is 3 amps?

So your mistake was in assuming that the total charge was 3 x the individual charge.