Funny how these stories transmogrify over time. I met and interviewed Mr. Moore in 1990 and he said Patrick Stewart who complained the "the Captain doesn't do enough screwing and shooting"...no mention of Gene.
I haven't been able to watch the video, but the transcript attributes those comments to Ira Steven Behr, not Ronald D. Moore. It also appears that Roddenberry was repeating his star's desires ("Patrick wants to get laid; he doesn't do enough fighting and fucking."). Since Moore wasn't in the room, and Behr also had a contentious meeting with Stewart after meeting with Roddenberry, I wonder if he just didn't misunderstand the facts at the time.
(On the other hand, it has been more than two decades. Do you have the old quote from Moore?)
Certainly in his new commentary on The Bonding Moore talks about Stewart telling him this, so in this video (which I can't watch either) it could well be Behr, or even something as simple as a brain fart on Moore's part.
On the other hand, there clearly was a push in the third season to try and placate Patrick Stewart and keep him on the show, so Gene could well have been encouraging writers to up the sex and violence for Picard as well.