As an an apartment brewer, I do see the genius in your process. I don't have a garden hose, nor a lawn with which to quickly hose out my MT. Subsequently, cleaning my MT is the single most onerous process that I do on each brew day because I have to do it in the bathtub/shower. It's the only part of brewing that I really despise.
I don't know what you're using as a mash tun, but here are some ideas from a person that has spent waaaaay too much time thinking about how to clean up a mash tun in a shower. 1) If you're using a sauce pan to shovel out your MT, you're doing it wrong--I did this for 15 years and it sucks!!! 2) If you're using a cooler, get a mash bag and use a drilled out cutting board to keep the mash bag from clogging your cooler's ball valve. Clean up is easy, just dump the mash bag into a trash bag, then rinse out the cooler. 2) If like me, you decided to get a serious MT that will allow you to do step mashes and recirculate, in my case, an SS Brewtech Infusion tun, just throw a trash bag over the MT, then invert it. This works well enough, but I really like your shop vac idea. A serious MT is a big, heavy, bulky thing that has handles and ball valves sticking out of it at very inconvenient angles. It's always an adventure trying to invert it into a trash bag without making a mess. By contrast, the base of a shop vac is, at least in part, designed to be emptied.
I like your idea! A thin plastic tub with wheels on the bottom is a lot easier to wash out when you're reaching into a shower than a 40lb stainless mashtun. I'm definitely going to give this a shot.
Thank you.