"Could the U.S. military really afford an invasion of Iran any time in the near future? We seem to be stretched thin..."
You are confusing "occupation" with "invasion."
It takes a lot of troops to occupy and pacify a hostile nation. It doesn't take a lot of our troops to smash a foreign nation and remove its leadership from power.
We can easily roll our Armies East from Iraq and West from Afghanistan and topple the Iranian regime. Should Iraq and/or Afghanistan need to be re-conquered and re-occupied, then we'd do that again later.
So we aren't stretched "thin" in regards to our defense...just in regards to occupation...and occupations aren't mandatory.
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Whether invasion without occupation would work in Iran is highly problematic, and I would say doubtful.
The student and freedom movement is strong, but they aren't the army and they don't have the guns.
Don't underestimate how much 25 years in power has gone to the heads of the clerics and imams who run the place either. They won't go back into a secular society without a great deal of resistance. We can thank Jimmy Carter for this mess.
The only way not doing an occupation would work is if there were an outside here who could be brought in--an even then it's a maybe.