Water turns cold after water heater circulation pump is off
Your expectation is correct, once the warm water is flushed you should then have hot.
There are 2 types of recirculating systems one one uses the cold line and has the pump and check valve (I believe this is the type you have) in a recirculating system on the cold line if the check valve fails it ends up being a mixing system.
A 3 line system has a separate line the hot is recirculated on the second line so it would not be long even with a failure until all the water in both lines was hot.
So check your pump system as I believe the valve is allowing the hot to mix with the cold while in use and that should not happen.