I don't see why you wouldn't be able to do it. So, I do not see a technical limitation.
However, using a direct method to solve (even a tridiagonal system) can be an overkill. You might be able to get away with only a few iterations (implying only a couple of very cheap matrix-vector products) to get a reasonable smoothing using an iterative method, while you would have to actually solve the system using the direct method.
Mathematically, I would express it, as follows. Define $t_\text{MVP}$ to be the time to perform a matrix-vector product, and $t_\text{Thomas}$ to solve a tridiagonal system with the same matrix.
For conventional workflow, you would require $t_\text{MVP}N_\text{iter}(\varepsilon)$ to achieve the desired accuracy $\varepsilon$. So, you would prefer to use the iterative technique when
$$
t_\text{MVP}N_\text{iter} < t_\text{Thomas}
$$
which would be true in a lot of cases. Here, $N_\text{iter}(\varepsilon)$ is the number of iterations to converge to the desired accuracy. For multigrid smoothers, $\varepsilon$ is usually not too high, thus making iterative solvers a general choice.
Moreover, the more complicated is the matrix, the more costly it is to perform a direct solve.