I think nearly all flowering plants do throw more stems when the lead one is cut. The only ones I can think of which this wouldn’t work on are things grown from bulbs or corms, as the flowering stems are formed the previous year.
Cosmos are a half hardy annual, so might not overwinter like a hardy annual. They will need to be kept frost free at a bare minimum. If that is going to be difficult, I would let them flower and enjoy them now.