I like the idea, but Lodge is the only of these that works well as a shelter IMO. You already said what's wrong with Acolyte's Abode, and Swamp Cabin seems pretty strong for a shelter. I guess it could be an interesting decision whether to keep it or not, but I think it potentially makes too much difference for a shelter.
Good point, somebody with 4/3 being able to leech off somebody elses's 5/2 isn't a good idea.
Here is an alternative for the Reaction-Shelter which is also more similar to Hovel's self-trashing:
I don't agree though that Swamp Cabin is problematic. Sure, it has nothing to do with the trash theme of Dark Ages but it is hard to come up with some novel on-trash bonus. Like Pasture, Swamp Cabin is something that would be dubious as Kingdom card so why not stick the idea on a starting card? I don't see its craziness given that Pasture is an automatic Duchy, often leads to 8VPs when you won the split (or even more in non-mirrors) and has a Kingdom card that supports the alt-VP strategy.
I also think that you will stick with Pasture more often than with Swamp Cabin.
On a sidenote, I think that in a world in which Pasture did not exist, a Shelter which would be worth 1VP per Estate you have would be good idea. Totally narrow but perhaps a thing in a Gardens or Silk Road Kingdom.
I don't like Heirlooms like Goat which speed up the game too much so I don't want to do anything crazy with Shelters but being an alt-VP enabler isn't something that messes with the opening. Here the seemingly innocent Lodge is the far worse offender as it makes terminal draw a pretty good opening buy due to zero chance of collision after the first shuffle (this is not the case with Necropolis).