Even if he's never thrown Hail Marry to end the half it certainly doesn't imply he will always check it down. Looking at his end of half opportunities ( Yes, I have too much time available) he's has had 2 realistic opportunities to throw a Hail Marry, to end the half. One of those opportunities was in 2015 against Tenn, we were up by 3 points at the end of the half. The other was against Buffalo this year in the rain, with a bad back.
He's had two more chances sort of but not really. In that blowout against the Rams a few years back he was at his own 48 with 12 seconds left. In that scenario you don't do a Hail Marry. If you take a sack you put the other team in FG range with time on the clock, or if it's incomplete and time is left they have their own chance.
Another chance was his 2nd ever game his rookie year but McFoot didn't get out of bounds and we ran out of time. We had 12 seconds left and were probably trying to get into FG range with his throw.
So I'm not exactly going to say he's always or is to chicken (conservative) to make that toss. You really have to be in a certain situation at the end of the half to make that call and throw. You have to be outside your kickers range but inside your QB's arm range. I found 2 such instances where it made sense to throw the Hail Marry.