From a service and safety perspective I much prefer the unit outdoors but I understand the "in the dry" convenience.
I've seen too many units with firewood packed so tight you couldn't replace a pump or fan. I rewire/plumb a couple burnt units per year due to spilled ashes inside a very dry shed. Cleaning a tall chimney plugged with creosote is a joy
...
Interestingly, it seems to me that units outside are cleaner because fly ash washes off instead of sitting on the roof.
Anyway, I do realize we don't have the snow down here to deal with like some of you guys do further north and it would look different in different situations. Climate is a big factor. Not trying to be overly negative, just emphasizing indoor installations can introduce unexpected things sometimes.
The floor of the unit should be insulated... As was mentioned it should dry out quickly and cladding should be designed to shed water. Maybe some caulk in some cracks might help.