Of course an independent Scotland would have to reapply for EU membership.
"Our people are already EU Citizens"? You only get that by holding a qualifying nationality. EU Citizenship has no independent existence. It consists solely in citizenship of a member-state, the definition of which is purely a matter for the member-states separately. Exchange a qualifying for a non-qualifying nationality, exchange British for Scottish Citizenship, and you lose your EU Citizenship, as surely as if you gave up your British passport on acquiring an American one.
"We have MEPs"? And once you had left, then you would no longer have them.
"Our students are away on the ERASMUS scheme"? And once you left, then they no longer would be.
So it goes on.
Who knew, eh?