It is not the case that the spin part of the state must be symmetric or antisymmetric. Instead, it is only required that the overall state is (in the case of electrons) antisymmetric. If the state is separable into spin and spatial parts (i.e. it can be expressed as $\lvert\chi\rangle\lvert\psi\rangle$, where $\lvert\chi\rangle$ is the spin part and $\lvert\psi\rangle$ is the spatial part), then it must be the case that of those two parts, one is symmetric, and the other is antisymmetric. However, it is also possible for the spin and spatial parts to be entangled, in which case the individual spin and spatial states do not need to be (anti)symmetric, as long as the corresponding state with the particles swapped also appears in the superposition, with the appropriate sign. For example, the state $\lvert\chi_1\rangle\lvert\psi_1\rangle-\lvert\chi_2\rangle\lvert\psi_2\rangle$, where $\lvert\chi_1\rangle+\lvert\chi_2\rangle$ and $\lvert\psi_1\rangle+\lvert\psi_2\rangle$ are both symmetric, is antisymmetric under exchange, even if none of $\lvert\chi_{1,2}\rangle$ or $\lvert\psi_{1,2}\rangle$ are.
Going back to your example, in saying that the state is $\lvert\uparrow\downarrow\rangle$, you are assuming some way of labelling the electrons. Since these particles are indistinguishable, we cannot say whether we measured the "first" or "second" electron; however we can meaningfully label them by their spatial states. For instance, if we say that the "first" electron is in the spatial state $\lvert\alpha\rangle$ and the "second" electron is in the spatial state $\lvert\beta\rangle$, then after measuring the spins, we would say that the state is $\lvert\uparrow\downarrow\rangle\lvert\alpha\beta\rangle-\lvert\downarrow\uparrow\rangle\lvert\beta\alpha\rangle$.
