Delhomme is tied for fourth in the league with 15 interceptions (Palmer/Cin 18, Testaverde/Dal 17, Plummer/Den 17, Bledsoe/Buf 15, Feeley/Mia 15, Collins/Oak 15), and last year was tied for seventh with 16 (so he's on pace to get more than last year). He's got good numbers this year with yards and decent connects for TDs but only his coach really knows how likely he is to give up a pick in the clutch.
A *short* pass to the sidelines that is intercepted is more difficult to recover from (tackling the interceptor) because of how few offensive-turned-defensive players you have between the interceptor and your own goal line. The interceptor is basically as close to the goal line as anyone because he is on a straight line to the end zone pylon and even your quarterback is on an angle to it, and he (more often than not) has momentum taking him in the direction of scoring while everyone on your now-defensive team has their momentum carrying them in the wrong direction (after the interception). Furthermore, because the interceptor his near the sideline, he's also as far as possible from any of your erstwhile tacklers (who, as members of the offensive team, are primarily not used to tackling anyone on a regular basis). Short passes to the sidelines during hurry-up play are both the most likely time for an interception and the worst possible time for it.
Sorry, I agree that it *might* be worth the risk but there's no denying that it *is* a risk (while you seem to be saying it is a no brainer and actually "stupid" to be doing otherwise).