Oh, I see. You actually have two slots, but only want to enable one drive or the other. It wasn't clear before. So you are going to use two of these adapters and on each cut the PERST trace between the edge fingers and the slot connector and patch in your switching circuit?
Your circuit as shown should work, but I am curious why you chose open drain output AND gates. If it were me I would use a push-pull '08 and a pull-down on the output to ensure default state of reset, just in case. It shouldn't need to be open-drain as there's only one device (the SSD) on the output, and there's no voltage translation needed.
If concerned about ESD from the switch (presumably mounted outside enclosure), diodes are good but I would also add a series resistor of a few hundred ohms (no more than 1k, otherwise logic low voltage level will be too high) between switch and diode for extra protection.
Where is the 3.3V sourced? Also from the slot? That would be preferable, as it ensures the switching logic is always powered up simultaneously with the drive, so there is no period of uncertain logic level on the output. You should make sure to have each AND gate (and pull-ups) powered only from its associated slot - i.e. not commoned - in case the system does something weird like switch power to slots individually.
I really don't think you need to worry about holding a drive in reset for lengthy indefinite periods. The drive won't know the difference between 2 seconds and 2 hours (or more). If it somehow does cause problems, then I would say the drive is badly designed.