Hi,
Yes, the tell-tale (pisser) stream should begin 2 or three seconds after start up.
You will need to remove the side cover to check and/or clear the nozzle, hose, fitting in engine and the passage that the fitting threads into. Small pebbles, twigs, grass and other debris can get trapped inside any one or more of those items and stop the flow.
It's something that happens to all of them from time to time and dealing with it is just part of being an outboard owner.
Some people use wire or weed trimmer string to push through the passage and sometimes compressed air will clear it too. I prefer to remove the fitting and start the engine briefly to allow the pump to push out whatever is in there. That, though, assuming that the water pump is fresh and powerful.
The little rubber hose that connects the engine fitting to the nozzle in the side cover gets pinched sometimes when someone routes it wrong or doesn't put the cover back correctly. It also gets old and hardens and needs to be replaced very few years because it will trap things more easily if it isn't soft.
Also, in your video I hear quite a bit of noise that sounds like the valves might need adjustment.....
.....just sayin'.
Good luck.
