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This is not normal, the "telltale" weep of the cooling flow is there to let you know that the pump is working. That flow should not be affected by the thermostat directly.
Assuming you are in Salt water:
Your tube to the telltale could be partially encrusted with salt, and as the water warms up expansion pinches off the flow (much as your bathroom taps "slow down" when hot water gets through). Also, perhaps with the thermostat closed initially there is enough pressure to push past the salt, and as the thermostat opens pressure to the weep drops and can't get past the blockage.
The loss of the weep telltale doesn't necessarily mean the impeller/pump is shot, but now you have no indication either way.
Remove the tube, ream/clean it out, clean out the port in the weep hole and check it again. It sounds like a fresh water flush wouldn't hurt either.