Starfleet is the Federations defense agency
I would agree.
Also that the Federation Starfleet has always existed.
Look at it this way, when a Turkish warship is the command ship of a NATO fleet (like off the horn of Africa), it is referred to as the NATO flagship, but it's still a Turkish warship. The other ship's in that fleet are referred to as NATO warships, but are still the property of their individual countries.
NATO flag being flown on a Dutch warship.