Would there be any disadvantage (other than a high component count) to using seven 470 μF, 16 V electrolytic capacitors instead of one 3300 μF, 16 V capacitor to smooth power supply ripple?
In general, the capacitor selection depends on the power supply impedance vs frequency that you must achieve.
In the specific case of 1 large electrolytic vs. several smaller ones, it is usually a concern of construction volume shape and BOM availability. Ripple current capacity and ESR tend to be rather similar, and ESL is usually not a concern for the frequency range addressed with the electrolytic capacitors.
It depends on the type of power supply and what the power supply requires from the capacitor to function properly and where in the power supply the capacitor is and what is the function of the capacitor.
If the ESR and ESL of the combined 7 caps are much lower compared to single cap then a regulator may not work properly if the ESR and ESL of output capacitor go out of stable range.
If the single cap was already a low ESR one then there might not be a huge difference.
7 caps may take more space and cost more which may be a disadvantage.