The engine kept running.
Earlier I had referred to the possibility of a failing ignition switch. Some Hondas are under a Recall for this but yours is not. No matter, the non-Recall Hondas suffer just as much as the Recalled ones from failed ignition switches.
Keep in mind that the main relay fails for the same reason the switches do. High electrical current over time due mostly to the fuel pump. The pump, main relay, and ignition switch are all lined up in a daisy chain. As the pump ages, fuel filters partially clog, etc the amount of current only goes up and with that current comes even more heat.
This is just my 2 cents so take it for what it’s worth. Those switch failures on Hondas are due to poor electrical engineering. The same poor engineering that caused 2 switches to fail in my old Mercury. Honda runs the fuel pump current directly through the ignition switch and Mercury/Ford did the same thing with the cabin blower motor. Heat and plastic do not work well together well over time.
In my 1988 Accord the ignition switch only carried the current for the relay coil, not the pump.
That is the whole point of the relay: to control a large current (through the contacts) with a small current (through the coil).
However, the ignition switch carried current for a number of other things.
Every Honda diagram I’ve ever looked at shows the fuel pump, ECM, and other things having their main power supply routed through the switch. The hot in run or start means it comes from the switch.
https://www.justanswer.com/honda/1fo44-1988-honda-accord-wont-start-found.html
Okay (4450) I stand corrected.