Wasn't Cosby's version promoted for prime access? It aired at 7:30 pm here, also against Jeopardy!, and IIRC, 6:30 pm on one of the Dallas stations.
IIRC, the Cosby version of YBYL was pitched as a competitor to the WOF/J block, and stations were encouraged to air the show in the 7-8 p.m. ET time slot. I was a freshman in college in DC when this aired -- it was originally at 7:30 p.m. on the CBS affiliate (WUSA), but was quietly shipped to overnights by early 1993.
Tried watching the Cosby version a few times -- it was just dull, and Cosby seemed disinterested and coasted throughout (There was a blurb in "Spy" magazine around the time of the show's cancellation that Cosby would pop in to the studio a few minutes before taping, essentially ignore everyone around him, and spent all of the downtime on stage reading newspapers -- supposedly had several of the day's papers under his desk/podium to peruse when the cameras were off).
As for Leno, I can see why he was picked for this show. He's a name and arguably one of the bigger "free agents" out there on the TV market. There is an audience, albeit one that is much older on the demographic spectrum, that miss him on late night.
And the Fox stations need something to air next fall. Several Fox O&Os, but not all, air "Judge Judy" (including Atlanta, Orlando and DC), and first-runs for that show end next year (reruns will continue, but it's expected they'll be shipped off to the MyNetwork or CW sister stations). That's an hour of programming right there, primarily in the 3-5 p.m. ET slots.
But I don't have confidence that this show will succeed. The game element is flimsy, and even his staunchest supporters will admit that Jay is not a great interviewer. He's going to need a lot of help -- particularly a strong writing staff and contestants that can interact well with him -- to make this work, but I have my doubts.