So my understanding is that while the way Lagrangians are written in classical theory doesn't extend directly to quantum mechanics, the concept of a Lagrangian is still useful since the Lagrangian can be fed into the path integral formulation (as opposed to being used as an input to the Euler-Lagrange equations). Also, in quantum field theory the starting point for canonical second quantization is typically a Lagrangian, where the fields are changed to operator fields.

Also, something interesting I came across - the Euler-Lagrange equations do have a quantum analogue as well: https://en.wikipedia.org/wiki/Schwinger%E2%80%93Dyson_equati...