Note that the reaction time is ponderously slow, barely enough to handle mains frequency rectification... those photo-MOS SSRs are quite slow indeed!
There are always two costs to a diode: the DC characteristics of forward drop and reverse leakage, and the dynamic characteristics of forward/reverse recovery.
You can have an extremely fast diode (no recovery), with low voltage drop, but you'll pay for it in reverse leakage and low breakdown voltage: schottky diodes.
You can have a low voltage drop (a "precision" rectifier), but you'll pay for it in recovery time, or auxiliary power (both as shown above!).
Note that a comparator takes time to decide, approximately inversely proportional to the magnitude of the input signal (plus a built-in minimum delay). This matters for very small inputs, on the order of a few mV. Most comparators go equally fast when the input changes by >50mV, whether it's 100mV or 10V: that's the minimum delay. For smaller signals, though, it takes extra time to... "think about it".
When using op-amps only (in a precision rectifier for signal purposes), the delay manifests as the amount of time required for the output to slew from saturation (or from one diode, if using the diode-wraparound-to-neg-input trick) up to the output voltage, plus a diode drop. An op-amp is more fundamentally an integrator, so that the output won't simply transition instantly: it always follows a slope, and that slope's maximum rate is limited by the input stage (which tops out when the input is over about +/-50mV).
Even very slow comparators or op-amps are faster than MOS SSRs, though. The SSRs turn on and off in milliseconds; comparators in microseconds.
What's worst is, because the comparator ideally switches at zero, it can only turn off the SSR once the voltage has fully reversed: and therefore, you're drawing negative current. Even if the SSR were instantaneous, the comparator still takes time to decide, and therefore limits your turn-off time.
This isn't meant to be an overly elaborate and harsh criticism of your circuit -- just to say that, there are many places where you can lose, and if you think about conventional diodes as having these elements inside them (how does an ordinary silicon P-N junction know to slew like a comparator?!), you should realize, it's
damned hard to beat them at their game! You can very easily come up with a circuit that wins in some respects (DC is easy!), and loses greatly in others (recovery time, voltage/current range, auxiliary power requirements..).
Active rectification is definitely a thing, today, but it's largely special-case: switching converters, where they can anticipate delays by generating a drive signal in advance; or low frequency rectifiers where the control circuitry doesn't need to draw much power (like
http://www.linear.com/docs/43273 ). With all our technology today, it's simply not possible to beat a UF4007 at its job!
Tim