> Previously I would just click "New Tor Circuit for this Site" a couple of times, and I would hit on a country relay that would allow me to reach the site that was blocked in another country.

What happens when you click on "New identity"?

I don't understand why you think (yes?) that the problem is not with what exit router your circuits are using (which could certainly lead to blockage, e.g. I often encounter blockage from UK sites when the exit router is not registered in the UK) but with the entry guard.