besides if you buy parts for that machine as "Deere" parts prices are likely to be higher than buying them as "Briggs" parts
looks like the old 391681 diaphragm which are as common as day around here. Should be able to get them in Stens or Prime line, even. (will likely have a different PN, but their catalog usually references the original Briggs PN. It'll tell which Briggs PN that it "replaces"
I did not look it up, just looks like the same one on the old 92000 series (3-1/2 HP) Briggs. That would be a Pulsa Jet carb (2 tubes that protrude into tank) vacu jets usually only had a single tube coming out of carb.
I remember back in my HS days I had a buddy that worked in a small engine shop--they were an Ariens and a Cub Cadet dealership. I remember him talking constantly about putting "warp kits" in that style of Briggs carbs/gas tanks. I have not ever done so myself but have heard that the gas tank flange on those, where the carb mounts would be out of square causing a vacuum leak. Not sure what those "warp kits" would consist of.
I do remember the 11000 series like that seemed common for valve seats coming loose. Didn't you have that on your Snapper?
I do remember having to peen a few of them in, back when I played with push mowers more than I do now a days.