This is not an answer but too long for a comment.

It was shown in

Popa, S. and Takesaki,M., The Topological Structure of the Unitary and Automorphism Groups of a Factor, Commun. Math. Phys. 155, 93-101 (1993)

that the unitary group $U(R)$ of the hyperfinite $II_1$-factor $R$ is contractible in the strong topology (which is equal to the strong-$*$-topology in this case).

Years before, it was shown in

Araki, H., Smith, M.-S.B. and Smith, L., On the homotopical significance of the type of von Neumann algebra factors, Commun. Math. Phys. 22, 71-88 (1971)

that the first homotopy group of $U(R)$ in the norm toplogy is isomorphic to $(\mathbb R,+)$. Under this isomorphism, the element $\lambda \in [0,1]$ is represented by the loop $$[0,1] \ni t \mapsto \exp(2\pi t p) \in U(R),$$ where $p\in R$ is some projection of trace $\lambda$.

This shows that the homotopy type depends heavily on the chosen topology and there is no reason (at least in general) that one should find an easy approximation argument.

This is not an answer but too long for a comment.

It was shown in

Popa, S. and Takesaki,M., The Topological Structure of the Unitary and Automorphism Groups of a Factor, Commun. Math. Phys. 155, 93-101 (1993)

that the unitary group $U(R)$ of the hyperfinite $II_1$-factor $R$ is contractible in the strong topology (which is equal to the strong-$*$-topology in this case).

Years before, it was shown in

Araki, H., Smith, M.-S.B. and Smith, L., On the homotopical significance of the type of von Neumann algebra factors, Commun. Math. Phys. 22, 71-88 (1971)

that the first homotopy group of $U(R)$ in the norm toplogy is isomorphic to $(\mathbb R,+)$. Under this isomorphism, the element $\lambda \in [0,1]$ is represented by the loop $$[0,1] \ni t \mapsto \exp(2\pi i\cdot t p) \in U(R),$$ where $p\in R$ is some projection of trace $\lambda$.

This shows that the homotopy type depends heavily on the chosen topology and there is no reason (at least in general) that one should find an easy approximation argument.