I chased a very similar problem on an old leaf blower once. And it turned out to be the main fuel line from the tank to the carb. When it was cold, the line was fine. But after the engine ran for a bit and everything got warm, the old rubber line would get a little too soft and could collapse...
Hey man, that's some creative thinking right there. I see what you're trying to do.
To answer your question, no, the engine vacuum won't open the choke for you. The choke plate is usually too heavy and isn't designed to be moved by air like that.
I've been down that road before, trying to make a...