In the meantime, any developer can make their own site more color-blind friendly without waiting for the drivers or the whole web to change. Why wait for someone else to solve a problem you can solve right now?
I don't think it's as easy as you make it sound. The hypothetical driver would need to map everythink produced for "standard" trichromatic vision "down" to dichromatic vision in a meaningful way, maintaining contrast where it is relevant, but how should the driver know? (Maybe I am overestimating the technical challenge, so please correct me if I am wrong)
I think the solution can only be to make sure semantically relevant differences in color (e.g. different positions in a chart) can be identified by people with dichromatic vision, e.g. by not using the popular disticton of light red vs. light green. Quickly viewing a design through one of those filter programs is no big tasks and should be part of basic accessibility checks.
This doesn't mean you can't use all kinds of colors when they are not relevant to understanding, [edit: removed direct reference to deleted parent content]
Oops i deleted the comment hastily. I don't see the reason why it's impossible to calibrate a mapping according to the specific user's color blindness. That should be enough to cover most cases, apart from the cases where you are actually asked things like "punch the red monkey"
A mapping of a 3 dimensional space to a 2 dimensional space will bring some points (colors) too close together, this happens with the missing color cones. So now the driver tries to correct this by changing the mapping from 3d to 2d color space, in order to move green and red (for example) apart in the eye of the color blind person. The likelihood that this mapping will bring together other colors, which before were meaningfully apart, is high.
The mapping could work like high dynamic range mappings, trying to take into account local image features but this would probably make everything look really garish :(
I agree that a switch/mode in those color-modifying lense programs could be to remap color channels so a color blind person trying to read a specific chart or website could temporarily remap colors in order to better tell apart the colors used.
edit: disclaimer: I don't know about anybody with limited color vision personally so I just try to be careful assuming the problem could be solved easily. If testing with actual people with this condition proves me wrong, great.