Key should be pushed in while cranking on cold start.------No it should not increase in revs up like that on start-up.----I would take the head off for a look see inside the motor.-----In the big picture a head gasket is cheap.
kdiddle------Please be more thorough in your research.-----The 28 in that 15-28 is a totally different motor than the 28 HP that you have.----Hurry and visit a boat shop for some help.----The internet searches are failing you !
The prop that shows up ( picture ) on this website is a through hub exhaust.----Not a pin drive either.-----There is no way you should be putting a 9" diameter prop on this motor.-----Hurry find a boat shop for some old time factual help.
Don't buy a prop for your motor through this website.-----The listing for 40 HP 1960 to 1976 shows totally wrong prop !!!----40 uses same prop as the 28 HP.
You need to learn that the water from the tell tale does not even circulate through the block or the thermostat !!----It tells you that the water pump is working.----It does NOT INDICATE that motor is cooling properly.
The tell tale should show water right after start-up.----Thermostat is not involved with tell tale.----Tell tale is plugged.----Or not enough water in your test bucket.----Or needs new impeller if it has not been done for 5 years.