I tried the suck/blow and when I blow there's no movement unless I push the valve down. When I suck there is no movement at all regardless of the button.
The diaphragm is soft, but it may be a little too loose. It has some imprint in it and it may be damaged as the previous owner had the diaphragm/metal insert order backwards. The smaller check valve is certainly an issue in my carb though. I thought it was seating well but when I took it apart again, it's not seating anywhere near the hole. This diaphragm is also harder, so there's not much movement.
The screen I cleaned before, so it's ok. After looking at the gaskets more, they're probably ok.