Your sentence above is correct, as would be the one with the structure you wrote between parentheses ("... than their secretaries do.") This sentence I just wrote is an example where inversion is not just a stylistic option but actually required. This would be very awkward: Your sentence above is correct, as the one with the structure you wrote between parentheses ("... than their secretaries do") would be.
In reply to your question, Is "than do" ever needed?, my answer is it is if the subject is particularly long, or when so recommended for stylistic reasons (which is frequently the case with comparative structures). It is of course also possible to omit "do": Hedge-fund managers who are pulling down billions sometimes pay a lower tax rate than their secretaries.
With short subjects like the one in your sentence, both positions are possible. Here's a similar example I found on COCA:
- Doctors usually know far more about medicine than do their patients.
And here are some others where inversion is clearly either necessary or advisable:
- Furthermore, farmers often believe that they experience far greater exposure to occupational hazards than do their employees, including farmworkers.
- Federal employees generally make approximately 20 percent more in salary and full compensation than do their counterparts in the private sector.
- All three Asian-American Christian groups attend services more frequently than do their counterparts in the general public.
- Low-expenditure households devote a smaller share of their budget to gasoline than do their counterparts in the middle of the expenditure distribution.