Being wheat intolerant in a way that's too specific to wheat to be gluten intolerant, more than half of the bread I eat (a slice most weeks) is 100% rye that's usually imported from Denmark or Germany. I rarely get the caraway flavored types. Get used to rye flavor on its own and I predict you'll like it.

Rye makes good American whiskey. On the other hand I'm not impressed with whiskeys made from only barley Scotch and Irish. Once I tried beer made from rye and it was nasty. So to me rye is for distilling, barley is for not distilling. At least there is symmetry.

Rye is very slow to rise because it has so little gluten. I take it your baking process includes forming the dough the day before so it has a full day to ferment and rise?