Yeah, it works fine. There are some application notes on S&H amplifiers - you will see this type of circuit. Two voltage followers buffering a switched capacitor. Switching analog signals with a 4066 is no problem. As far as I know, the only time you need a "true" analog switch IC is if you're dealing with varying voltage levels for signal vs control logic, as those ICs offer a separate logic supply.
Actually, you can buy this very same circuit in a single IC, for example the LF398.
If you choose the LM324 you could try it without any virtual ground, but there would be a little bit of offset error near ground. If you do use a virtual ground I think it would be a good idea to put a buffer after the divider.
Nice! Look up the TI app note on sample and hold amplifiers (I'm on my phone now or I'd link you) and try out the variation called the "closed loop" configuration. I think it may help with your stability issue. -jkc
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