RB, you expressed my experience there perfectly.
Did you by chance "wander" as far as the other side of I-75 to give Jet Blue Park a try? I'm thinking no, because if you had, you would know the answer to your question about whether the open experience is available to all other fans: the answer is, no. A lot of their back fields are roped off, or at least were the couple of years I tried to go see. It's not bad there, but nothing like at our Twins' facility.
If I am at Ft Myers with no one else, I might not even set foot inside the ballpark for the nominal "major league" games. You can see major leaguers on the back fields anyway, now and then, especially on days the big club is at an away game. Watch Byron Buxton bat leadoff for each of the first four innings, followed by Carlos Correa (or maybe it was Jorge Polanco)? Heaven. I hadn't sat so close to Byron since that one time back in the Arizona Fall League.
If anyone is on the fence about going (aside from money), I have two words: Do It.