The base current has to come from somewhere, namely the power supply, so you get a little ding on the rail. Since you are driving a cap, you are driving a load, thus you are using the output stage and the low beta does effect the operation. It depends on the quality you are trying to achieve.
Regarding something being unstable, you need to consider power up. That is, can you be "born" in a mode where the thing just latches. There are two scenarios of concern. One is the slowly rising power supply, typical of some DC/DCs. Analog circuits can behave differently depending on how they are started. Then there is the scenario most people don't care about, i.e. the thing is running and some asshole quickly kills then applies the power. That happens with power supply glitches, or in the case of a chip, some oaf shorts a pin to ground or a supply when poking around with a scope probe. [The power glitch is easily solved with reversed bias diodes to the positive rail going to the cap. That is, if the power is cut, the cap is pulled back to ground with the diode. Often this is free in CMOS design since you have parasitic diodes there anywhere.]
It all depends on how much you value return customers. Some people simply accept that things need booting. Others put devices in remote locations and are pissed if your thing needs a kick in the ass. You don't get the next contract.
The other thing to consider with op amps versus comparators is that linearity can be your enemy. In chip design, where a gate or two is nothing, you often see comparators followed by a logic buffer. One is to insure there is no fanout issue, but mostly you want to limit the amount of time a signal is in an undefined state, i.e. switching. But this is true of board level design as well, just that the extra gates are not so free. The extra gates cause delay, so you can't go overboard either. I had a situation where a multivibrator was getting stuck due to the "inverter gain" dropping. I never did figure that out, but I had extra inverters there just in case and made a metal mask change, adding the extra inverters, which solved the problem.
Any good application engineer will test power supply or ground shorts to all pins on the chip. It doesn't mean you won't sell a chip that fails a short test, but they have to know how the chip behaves under all conditions. On board level, maybe nobody cares.
The customer will never blame themselves for odd operation. They always blame the chip. You really need to idiot proof this stuff.