The part you quoted (to exercise like Authority over all Places purchased by the Consent of the Legislature of the State in which the Same shall be,) refers directly and only to those uses listed in Art 1, Sect 8, Item 17.
The fact that the fed bought the land from a foreign power, rather than gained the land by conquest, has no bearing on this issue, what-so-ever.
The uses for which the fed gov can own land are limited to only those uses listed.
You keep getting hung up on the “bought it” angle. That means nothing other than the lands were bought in order that they’d (the lands) be incorporated into the US of A, first as territories which the fed gov does manage directly, then, once the proper hoops were jumped through, as states. Once states, only those items listed can be owned as property by the fed gov.
I think we're down to the core issue of our disagreement. Where does the Constitution or any SCOTUS decision state, imply, or suggest that all land acquired by the federal government be used only for the purposes listed?
The Constitution grants specific powers to the federal government and it prohibits specific powers to the states. Everything else is reserved for the states.
Nevada disclaimed all right to the land (and authority over it) when it became a state. It gave authority over that land to the federal government and, to my knowledge, it has not since claimed any authority over the land. Are you suggesting that Nevada did/does not have that right? (Not argumentative. Sincere question.)