As I see it you have raised three issues. One is the FMLA which you have handled and I assume you are satisfied with.
The second is a payroll issue which comes down to a matter of your policy. Deviating from your business practice comes with a different set of problems especially if others want to follow and do the same thing. You have stated that you have been down that road and it has not worked very well in the past. Could you do it? I suppose so, but I would be concerned for the reason that you stated for doing it.
That is the third thing, confidentiality. If the ee is out on FML then you already have the specific reason listed in the FML Medical Certification if you are using the most recent DOL form. How would that change if the ee dealt directly with the insurer? Further, the company has a responsibility and legal obligation for medical confidentiality, which I am sure you keep very well. Recognizing the reticence on the ee's part is a compassionate concern but it should not replace the company's responsibility or rights toward FMLA, your payroll or the way you deal with the insurer. I would also be concerned for the bigger picture of the other employees who may want to deviate because inadvertently you have allowed them to distrust the company.
I am just one head (who handles FMLA for my company) opining on your question. Could be that others will feel entirely different. I await my most worthy colleagues comments.