As your picture shows a Winbond 25Q256FV presumably the same issue of the bootloader misusing the flash chip could exist, and would probably be fixed in the same way as for the AI7688, which is to say with a build of U-Boot the only ever operates the flash in the mode which the MT7688 is strapped to expect at boot..
In terms of having a serial line held high when the module is unpowered causing problems, indeed it can. More importantly it is probably outside the allowed electrical specification - generally I/O lines are not allowed to have voltages above the corresponding I/O supply rail, as this puts stress on the protection diodes. A series resistor on the serial receive line will help with the stress, but even then you can see reset issues.
It's best not to connect the serial before powering the module. If you want to have a chance to interrupt stock U-Boot via a character on the serial before it times out, power on with the reset asserted until you have your serial terminal connected and ready to press they "any" key.
If you really want to solve it with circuit, a several K series resistor on the serial line to limit the current, then a schottky diode to the supply rail to divert it, and finally a few hundred ohm resistor from the supply rail to ground will solve the problem by preventing the leakage current from raising the supply rail to the threshold voltage where it starts to prevent clean reset.