I appreciate all the union-related comments. But Desi had his own band on studio payroll. Now, they either sat around all the time until they played cues 15 minutes a week for the latest "Lucy" sit-com, or they were used all over the studio.
Plausible, but it's worth pointing out that prior to 1966, Desilu didn't
have any shows in regular production other than
The Lucy Show. It stayed in business by renting its production facilities out to other studios. So from 1962-66, the sitcom was pretty much the only Desilu show they could've worked on. (And I'm sure a scoring session for a half-hour sitcom takes considerably more than 15 minutes.) I'm not sure how that affects your assertion, but there it is.