Water heater is still good and supplying hot water. The whole system has been working fine for years. No hot water will come from any tap, however; I noticed when I had the bathtub faucet on AND the sink faucet on, only then would the water become hot. Blew my mind. Anyone know what would cause this?

Is the water heater tankless?

yeah, tankless heaters have an internal flow switch that turns on the gas or electric when it senses movement of water thru the coil.
if you have a tank type water heater with a recirculating line then most likely the check valve is the problem.

Check your Users manual. All On-Demand water heaters require periodic maintenance. I had a Bosch that you were supposed to remove the flow valve annually and clean and replace o-rings (Didn't know that. At 1 year and 8 months it quit completely. Set screw for flow valve frozen. Couldn't remove. $599 water heater became large boat anchor.) off. My current Rinnai tankless requires that you run vinegar thru the system for one hour annually. It has bypass valves and ports built in for this.

not tankless

It's a standard gas water heater, still looks in good shape and does produce hot water. There is an expansion tank and also there is a mixing valve with a thermostat (watts LF70A) that I am suspicious of but don't understand it enough to know how that would cause this if it is malfunctioning.

yes, it very well could cause your problem

if you are somewhat mechanically inclined get a 'rebuild kit' for the mixing valve and have at it

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"Measure Twice & Cut Once" - Retired U.A. Local 1 & 638

Thanks all. I will try a rebuild kit.