LSU and NFL caliber QBs. Why the void?

Just in the past decade we have had players from LSU who became immediate starters at RB, OL, WR, DL, LB, DB, punters. When was the last time LSU had a QB who was a regular starter in the NFL. My question is, why can't we recruit the QB position as well as the rest of the positions. I realize it is only one person/one position, but it's not like it would be a fluke for LSU to have an NFL-caliber QB. Two in 50 years.

Cheat sheet. From the LSU Media guide, here are the QBs who were even drafted in the past 50 years. Only two have been weekly starters:

Matt Flynn
Rohan Davey
Josh Booty
Alan Risher
Tommy Hodson
Jeff Wickersham
David Woodley (starter)
Bert Jones (starter)
Buddy Lee
Pat Screen
Jimmy Field

For that matter, on the all-time list, only Y. A. Title and Bert Jones were year-after-year starters