My house originally had a recirculating pump for the hot water but it burned out. Somehow though, it still (mostly) worked and had instant _warm_ water.

I think it was through natural convection/circulation - the hot water expanded in the tank and pushed it through the recirculating loop?

So maybe there's a good-enough solution that doesn't require a pump, just a return loop.

Now I have an on-demand water heater with a built-in recirculating pump, so it's instantly hot :)