The modules list will say error if there’s no internet connection. I think this has to be something on your router’s end that is stopping it. You could try bypassing the router and plugging the pi directly to the modem you use over rj45 and then connecting to the pi’s wifi with a phone or something to start installing modules.
Is it possible you’re turning off the Pi too quickly after shutting down? The best way to shut down is to go to the admin interface and hit the shut down button and then wait about 30 seconds. On the pi itself you will see it blink the green light about 12 times and then it will be a solid green for about 2 seconds. After that solid green light it’s safe to shut down the pi. MicroSD cards are known to corrupt easily when the power is turned off and is one of the benefits of using a USB flash drive instead, thought it can still happen.
You can turn off Etcher verification. I usually do as it takes too long, but it is nice to know that it worked properly when you’re having problems. If this is not a power down issue, and you have new MicroSD cards, the only other things I can think of is it being the MicroSD card reader you’re using to flash the cards or maybe the pi power supply you’re using for some reason. But that doesn’t make sense if they boot once.