If you just have a 6 or 10 gallon typical RV gas water heater, I am wondering why you would leave it on for long periods (like a standing home tank water heater.).....unless you have a large group taking sequential showers etc.? I think typical use is to turn it on about 15 minutes prior to need and then turn off again after the need is completed to save gas and wear and tear on the appliance......but each to his/her own of course.
From your description it sounds like it works fine (starts and hot water) for the cold start situations and only fails (or locks out) for the long term “left on” operation. Maybe what happens is that when you leave it on for long periods, the water heater cycles as normal to maintain water temp (low point for reheat is approx 115F I think), but perhaps it fails to restart once at some point (for some reason) and the control board locks out. It probably will not re-try to start until you manually turn off the water heater (on/off switch), which clears the lock out and it starts the heating cycle again. The one time or random “fail to start” situation could be caused by many things.....including control board, loose or bad connections, flame sensor adjustments; old gas valve, etc.
If your thermo and ECO sensors are not solidly mounted in place (especially thermo) that could result in erroneous temperature levels in the hot water. I would “guess” though if the thermo was not touching the tank it might run the burner longer than normal to reach the typical 140F and you would potentially get hotter water than normal. It does not hurt to change the sensors because I think you also get a new retention seal cover etc to hold them solidly in place. You definitely want the ECO and thermal cut off device working as designed.
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