Single-IC solution to the problem of spurious high on NodeMCU powerup
Your 7408-based solution sounds fine: just connect one input of all the AND gates to the common enable pin.

Other than that, there's a lot of 74xx series octal buffers with three-state output and an enable pin.  the Cheapest I found on digikey is the 74LCX541FT, which has active-low enable pins, but that's not really a problem: pull them up with a resistor to Vcc, and pull them down using your microcontroller once everything is stable.

Also, since relay boards are incredibly slow in comparison to microcontrollers: The most straightforward way might be using a serial-in, parallel out shift register. That reduces the amount of NodeMCU pads needed to control your relays from 8 data pins + 1 enable to 1 clock + 1 data (+ 1 latch pin, if you're using one that doesn't automatically only latch after 8 data bits). I²C IO extenders would be another solution.

The beauty in reducing pin count is not only reduced wiring complexity, but also that you could very well opto-isolate these 2 to 3 lines; a feature that can be desirable, if the relay switching leads to any transients that you don't want to expose your nodeMCU to.