If that wasn't you coughing or sneezing at about 14 seconds into the video then that sounds like it was backfiring, or trying to, out of the intake. A sort of classic indication of a lean condition or valve timing being off or both.
Are you in the habit of running the outboard fuel system dry when parking the boat? If so, the way the engine reacts might make this familiar to you.
That video reminds me of what many engines will do as the fuel in the carb goes low and the engine starves out and then quits. Only in this case, it picks back up like it's getting fuel again.
Have you checked the the things we discussed earlier? Tried tapping the float bowl while this is occurring?
I'll be honest here and tell you that I have very little trust of most shop's ability to get these little finicky carbs set right. There are many little things that NEED to be done. I know, I've done many of these and struggled with that myself in the beginning. A very WISE Honda tech talked me into buying the HONDA MARINE CARBURETION MANUAL and I sure wish I had had one from the start. Most shops don't have that crucial information and that turns cleaning the little critters into a game of trial and error or a "wash, rinse, repeat" scenario.
Having said that, the fact that it starts easy and seems to idle ok would indicate that they got it at least mostly right and I still think that it's either pulling in air or losing prime momentarily for some reason. But if you can't find a fuel FEED problem then it might very well be a carb issue that I've just never encountered before.
Good luck.