I asked Perplexity AI, and it said yes, people do work for both the Gates Foundation and private companies concurrently. (The Foundation even bought an interest in a Danish-American drug company, by the way.) I swear to God I did not bring up GSK in my question, but note the answer:

"Yes, there are instances where individuals employed by or associated with the Bill & Melinda Gates Foundation also have connections or concurrent roles in private pharmaceutical companies.
"The Gates Foundation often collaborates with pharmaceutical companies, and it is not unusual for staff members to be former employees of these companies. For example, the foundation has a tendency to hire individuals who are or have been in key positions in both the private and public sectors. Many staff members and advisory panel experts have backgrounds in pharmaceutical companies such as GlaxoSmithKline, and some hold positions simultaneously in both sectors4.
"Additionally, the foundation's partnerships with pharmaceutical companies involve close cooperation, which can lead to the exchange of personnel and expertise between these organizations."

So Rodolfo, please take the fud elsewhere. Thanks in advance.