Went through the same thing this past weekend on a friends 50 Suzuki, trying to install a water pump kit near the dock because we didn't want to rig down his sailboat. Six of six tips were missing or worn down from the water pump impellor when we removed the old one. I got impatient and put the new water pump kit in without trying to find the missing tips. As expected, it didn't get water out of the tattle tale. Took it apart, tested the pump with the lower unit off the boat and got good water flow. All the while exhaust was coming out the tattle tale which is expected. If there isn't water getting to the tattle tale, there's nothing to prevent exhaust from flowing backwards in the cooling water system, even out of the tattle tale. Tried shoving a coat hanger wire up the water tube and felt resistance before hitting metal. Parts diagram showed there shouldn't be anything. Finally got access to a water hose, blew backwards through the tattle tale and heard good water flow through the head & exhaust but only a trickle coming out the water tube. Used the coat hanger to probe the blockage while back flowing water through the tattle tale and 5 impellor tips finally came out. Curved the end of the coat hanger to try to fish out the sixth piece and never found it. Put it back together and immediately had good flow through the tattle tale, for a while. Several minutes later weak or no flow through the tattle tale but the engine wasn't overheating. Pulled the hose off the thermostat housing and had good flow. The root cause was scaling on the block cooling jacket came loose when we didn't have water circulating so the metal expanded at a higher rate than the scale and it was flaking off. Removed the tattle tale several times and probed the socket for it and it finally quit plugging after we worked out all the lose scale.
As for a potential head gasket leak, if it's 4 stroke and leaking bad enough to prevent water from getting to the jacket, you should be having rough idle or idle at all as it would want to pull water into the cylinder on the intake stroke. One way to verify is to pull the thermostat out the leave the housing off while running the engine and see if water flows out. The pump is designed to overcome exhaust pressure so whether it's coming from the exhaust port or combination of exhaust port & blown gasket it can only flow the amount of gasket generated by combustion and the cylinder displacement less some thermal expansion absorbed by the cooling water.
