@StarTech - you've solved the mystery of why the carb looks wrong - I wasn't noticing my TB304H is a variant - 41BD instead of the 41AD that seems to be what all the parts suppliers show parts for.
It doesn't have the third line like the one pictured but the Troy-Bilt parts finder it says the 751-16046-4 is the replacement. Maybe the photo is wrong?
This Troy-Bilt TB304H is 2 years old. I figured rather than troubleshoot I'll just buy a replacement carb but I can't find one that says it is the correct replacement for the TB304H. The unit wouldn't run at all so I decided to clean what I could on the carb without pulling the diaphragm off...