As long as the "doghouse" provided protection from the elements, I see no need for the equipment to be outdoor-rated.
That said, maintenance is obviously an issue. Without the enclosure, the equipment needs to be at least Nema-3R.
I see many outdoor installs where the equipment is not so rated. Theses are invariably old enough that I wonder if there were NEMA standards at the time. That, however, is not the case here.
I say this, because the one disconnect has provision for slipping a padlock into place to lock it "off." This provision didn't become common until well into the '70's- quite some time after both NEMA and UL had standards for outdoor equipment. The motor starter also appears, by the lack of rust, to be much newer than the disconnect.
I am beginning to suspect that this instal was re-done, well after the doghouse disintegrated, and as such should be weather-proof.