Exactly. To be entirely honest I do not see the need for individual planetary fleets (or individual planetary governments for that matter, but that’s a whole different can of worms)
You need a lower level government for the same reason your country's head of government doesn't pull double duty as your mayor. The two jobs have vastly different scopes. The small stuff (e.g. paving roads) will never get done when the executive is busy trying to avoid a war. Attention, even a government's, is finite.
By the same token, local units don't have an incentive to move resources out of their turf. Starfleet has an immense territory. Unfathomably massive. They're in charge of defending the border, exploring beyond it, defending allies, and the interstellar space between systems. There's too much incentive to move resources around as they are needed. Entire sectors could find themselves drained of ships, which you probably saw during the Dominion War and the rushed response to the Borg Invasions.
Anarchy grows in a vacuum if there is truly nothing to fill it. That's where a local defense (which would be akin to a federal police, coast guard, or gendarmerie) come in. The local defense forces are, to borrow from yet another era, garrisson forces. They are equipped to defend the castle (for a time). They cannot sortie out on offensive or missions and are certainly not equipped for seiges.
A planetary defense force would not be equipped for long range deep space combat for extended periods of time, though some planets (Vulcan, Andoria, etc) might still have deep space cruisers left over from their independence for a while. Most ships would probably top out somewhere between a runabout and a Defiant class in firepower, accommodations and endurance.
The designs could be Starfleet, or local. It would depend on the needs and shipbuilding capability of the planet in question.
As for the Vulcan Science Academy, it appears to be a civilian entity (it may or may not have government backing after admission to the Federation, if it didn't it would explain the persistent and open racism, which is unlikely to be tolerated in a multicultural alliance like the Fed). They'd be free to build or purchase whatever ship they want.
