It's a forced lynch pin because it's an easy pill to swallow if you aren't so emotionally caught up in the potential ramifications.

There are a lot of plausible ways this could work, we really don't know enough about Westerosi laws and customs to definitively say Rhaegar could or couldn't do this. Taking real world examples, we do know that it was not at all uncommon for Royalty to pull stuff like this, of course with varying degrees of difficulty depending on the era, specific place, and specific circumstances.

It's a forced lynch pin of Jon's (potential) story arc because we simply don't know enough. If it is in fact the weak link that's holding this all together, then you can dismiss all the things it supports-- the things which don't conform to your idea of the story they should be telling.

Like so many other things, I think the books (if they go in this same direction) will easily sell this as convincing, however the show probably just doesn't have the screen real estate to spare. If that's the case, I'm happy that they're doing their best to adhere to the overall vision of GRRM, even if it means taking a few shortcuts and stretching our suspension of disbelief a bit to get there.