Well, the water not coming from the Tstat area bothers me but I don't know how you tested that. Were you using muffs or in a tank (trashcan) of water?
It sounds like the pump may not be priming. If you're in a tank, make sure the water is above the vent plate 2 or three inches. If on muffs, there should be adequate water pressure and good sealing cups.
The tell tale is best cleared by removing the hose fitting from the block and running in a tank. A good pump is VERY powerful and can usually force any obstruction out of the hole. Always makes a mess and can hurt you so be careful. You and anything nearby WILL get wet if the pump is good.
If you just slapped a new impeller into an old liner (cup) and housing and set it on a worn cover (wiper plate), the pump will probably be weak.Those parts need to be in good shape to work together. A new impeller can quickly fail if paired with worn parts.
Another possibility on an outboard this old is the drive key cutout in the vertical shaft can be worn and let the key get spit out. No key, no pumppy.
Hopefully there's not an old piece of impeller stuck up inside someplace.
The tell tale bypass is near where the water leaves the tube and enters the bottom of the engine block. If there's an impeller chunk stuck right there, you may be able to pick it out with the water tube removed.
Just a few ideas about what you may be up against
Good luck.