You really wouldn't want that, actually -- at 1MHz, you'll get all sorts of nasty peak voltages and currents from the unavoidable reactances (inductance and capacitance) present in the surrounding wiring. And coming so frequently that the thing will destroy itself. If it survives the first cycle, even. You really need extra circuitry around the switching device to limit that, and that circuitry tends to make it special-purpose. So that's basically why we still have bare transistors around, for the most part.
Note there are basically two kinds of SSRs: AC-only (thyristor based), and AC/DC (MOSFET(s) + photovoltaic driver). Thyristors (namely TRIAC or SCRs) turn on reasonably fast (fractional microsecond, which can be tricky concerning peak currents through capacitors), but stay on until current drops to zero, so can only be used on AC. The PV driver delivers very little current (microamperes), so turns on the MOSFETs very slowly (several milliseconds). The switching loss can therefore be pretty high, but that just means you don't want to turn it on and off very frequently (which is usually fine for such applications).
The MOS SSRs are also fairly useless at delivering fault current, so, won't clear a fuse before blowing up themselves. Thyristors can survive such rough treatment, just barely, if the fuse blows quickly.
Tim