I'm not sure this was invariably the case. The Beatles meant nothing on the R&B singles chart (as opposed to the Stones, who did have a few R&B hits). So when musicians like Willis Jackson whose albums were definitely aimed at the black community, recorded their songs, it doesn't seem to me like a good strategy for cashing in; you don't cash in on something that's unsuccessful.
MG
It's possible that Willis Jackson or others who appealed to a black audience didn't have a primary interest in recording Beatles songs, but perhaps their record companies did. Even then, rock (and funk) had put a devastating hit on the jazz market and everyone was looking for a crossover success, however it might happen.
You may be right that it was the producers who got these songs recorded by Gator and others. But these musicians had always been interested in playing white pop songs like 'Glory of love', 'Angel eyes', 'She's funny that way' and 'South of the border' etc etc. I saw the Beatles songs as being part of that thread of entertainment jazz.
MG
You may be right also. The only way to know would be to ask any of the artists or producers who are still around and hope for a straight answer.