Why does a Tesla coil secondary need intermittent excitation to achieve resonance?
The secondary doesn't need to be excited intermittently.
The second, less important reason is that for large coils, the peak power required during the excitation is so high that it would be a practical impossibility to supply that power continuously from a standard wall socket.
The main reason is that the original coils were designed a long time ago, when the only available power switch was a spark gap. This needs a steady build up of voltage, storing energy in a suitable capacitor, until the voltage becomes so high that the switch breaks over, dumping most of its energy into the primary coil. This automatically means intermittent operation. The most convenient voltage source that had this 'steady build up' sort of characteristic was AC mains.
A modification of the spark gap was the 'rotary spark gap', where electrodes were moved to modulate the breakdown voltage.
With the advent of power electronics such as high voltage IGBTs, 'solid state' Tesla coils can be run continuously, but only if they are small ones. If you do the sums for even a fairly modest coil in the 1m height region, the power required to excite the secondary can run into 10s of kW, which means intermittent operation is required for domestic use. You can make a benefit out of this drawback by timing the bursts of operation from a music input, to make a musical coil.