I'm not sure that it necessarily follows from the overlap they are not linearly independent. As far as I can tell that alone would simply make them non-orthogonal basis vectors.

The issue is more that there isn't really any frequency that only stimulates the M cones. There are some that more or less only stimulate the L cones, but as we get bluer, the stimulation curves of the M and L cones mostly converge, so we can't really get fully independent M cone stimulation (at least not without tricks like locating specific cones and using lasers to stimulate just the M ones, while missing the L ones.) This does indeed make them not linearly independent, so your overall point still stands.