Solved

Doctors are useful ?

The Doctor will make Driver health decline less each time the training button is pressed. High level Doctors will therefore allow you to do an extra training session each day and as a result you train your drivers more quickly.

Once a driver is fully trained (30 in all trainable attributes) then a high level Doctor has no benefit and can be replaced with low level Doctor which reduces your wage bill. If the Doctor has many races remaining on their contract, replace with a low level from the market, if only a few races remaining on contract then allow the contract to expire and you will be assigned a Doc by the game with wages of £50k per race.

As for the original question... When a driver has red stars they remain trainable, it is just very slow, 3% progress for every session. When your team levels up the stars will turn yellow again and the training progress will increase.
The downside is that as your team levels up the Doctor star rating also reduces and as a result they become less effective.

Bottom line... If your driver is still trainable AND you can afford to retain a high level Doctor then your training will be marginally faster. If you're struggling to balance game cash then the Doctor is probably the first place to make savings.