If by diaphragm you mean the air-box than you should either replace it or batch it up.
I'm going to guess that the engine is running lean due to carb rebuild or the diaphragm leak so pull out a spark plug to check how much air is getting thru without enough fuel, the tip of the plug should have a tannish color to it, grayish white is too lean, blackish is too rich.
Hmm... Sounds like you've got a project bike for sure! On why it's dies with throttle I'm not sure. Silly, but make sure the choke is in. If your diaphragm has a tear then replace it. Especially that big of a tear. Hope you get it running good!
Sounds like the common problem of a clogged Pilot Jet and/or passage. Had the same problem with my dual-sport, would start up fine but with any throttle used it would die out straight away. Had to remove float bowl and blow air down the passage that the Pilot Jet is removed from. Then blow air through the Pilot Jet to. When re-assembled, worked fine.
Replace the part that is torn. Sounds like a major intake leak.