So to the extent people are placing Amherst last because they don’t think a non-traditional transfer student would like it, I personally suspect that is not a valid assumption. Again, it is in a great area overall with tons to do, and Amherst, MA itself is a nice college town which also hosts a full research university with plenty of postgrads.
So Amherst the college would not define the limits of your non-academic life. It would be where you would be doing your academic work, but the rest would be up to you.
Otherwise, I am just going off what you have said.
I totally get the appeal of Georgetown given your interests, it could not be more plugged into that IR/NatSec world, so if that is your #1 choice if actually admitted, makes sense to me.
If you don’t get admitted to Georgetown then what should be your #2? Well, as you point out you would have to spend less time at Amherst, and specifically would not have to do a bunch of intro classes unrelated to your interests to graduate, instead you could go straight to taking the higher-level classes that interested you. Technically Brown would allow that too, but they are really going to encourage you to explore anyway.
Of course if you actually really wanted that full four-year general education before you went on to grad school, you could choose Yale or Brown. But it doesn’t seem to me like you do, so why choose them over Amherst?
Finally, Amherst is not somehow less suitable for grad school placement than Yale or Brown. You seemed originally concerned that Yale and Brown being Ivies was a big deal, but it really is not. So while they are among the many good choices for people with your interests, they are not somehow radically better than Amherst! Indeed, if you go on to a grad program, no one will particularly care where you went to college.
So my feeling is other people are treating this like a normal college choice question, except you are older. But I think you have an opportunity here to really finish college on your own terms, then go on to grad school on your own terms. And when I think about your options and what YOU want to do, it seems clear your favorite option is Georgetown, but your second favorite is Amherst and then a quick move on to grad school.
And that makes perfect sense to me.