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I'm not sure what kind of answer you're looking for since the question is very subjective, but I'd say yes, this feature is reasonably common.  I'm mostly familiar with Microchip PIC microcontrollers, and many of them have "dual speed startup" options.

Of course it's hard to tell from that how common it is that it actually gets used, but at least one major company thinks the feature is important enough to include in many of their parts.

I don't remember ever having started a microcontroller with the built-in RC oscillator, then switching to a crystal later if high timing accuracy is needed.  I may have started with the RC oscillator only to get fast wakeup response, then switching to the crystal when it was ready.

I have done something sortof similar with a PIC 10F where I used the watchdog timer to wake up the processor periodically.  It needed to be asleep most of the time due to running on a small battery. The processor would always run from the internal RC oscillator, but timing was lost between wakeups and I needed to keep track of elapsed real time.  To get around that, every 256 wakeups I'd leave the processor running to measure the time to the next watchdog event.  That effectively calibrated the time between watchdog wakeups to allow tracking total elapsed time well enough.