Sure, and I do that occasionally, but I should note that just about all of the food champions I have at the moment are fully leveled and waiting to be ranked up. Maybe about 1/3rd of the 3* ones are not. I rank up 1* and 2* and then use them to feed the 3* and then use those to feed the 4* etc. It's also worth noting that I don't usually have this many food champions active at the same time, but I needed to farm up shards and had like a full week of double XP with nothing else to do, so it ended up that way.
Either way, I'd much rather not have to use the inbox. In fact, I'd like a 3rd storage area, essentially a trash bin where I can put champions I know I don't want to keep and can safely use as food. Keeping them in the active list or the vault clutters the UI up for an area where I'd ideally like to keep the champions I actually use or intend to use at some point. Another solution would be to allow you to convert food champions to chickens so you don't have to store them anywhere at all, but apparently that idea has already been rejected.