Going back to 2002 in the FBG Data Dominator Tool, I have selected for a number of variations on Team Stats of passing attempts on 4th down of 4th quarters when down by 4 or more points. I looked at conversions divided by attempts. I did not include rushing stats at this time and that is a weakness, because even though they should all be passing, a QB scramble would count as a rush attempt and I can't really parse that. But this is adequate for a rough estimate IMO.
4th and between 5 and 15 yards to go from OPP20 to OPP-GL, and any score:
137/414 attempts 33.1% 87 TDs
4th and between 7 and 13 yards to go from OPP15 to OPP-GL, and any score:
48/155 attempts 31.0% 38 TDs
4th and between 5 and 11 yards to go from OPP15 to OPP-GL and any score:
99/278 attempts 35.6% 72 TDs
These are the closest to reality, though:
4th and between 6 and 10 yards to go from OPP10 to OPP-GL and any score:
36/113 attempts 31.9% 35 TDs
4th and between 6 and 10 yards to go from OPP10 to OPP-GL
and down by 4 or more:
35/111 attempts 31.5% 34 TDs
I started this OP going with a 40% number and this would seem to not be too far off from what the data actually show.
One wrinkle, though, is that Rodgers is inarguably the very best in the league at generating free plays and drawing an offside. In fact, check out the graphic on this link. It includes all downs of course, but it is a strong indicator.
https://operations.nfl.com/gameday/analytics/stats-articles/aaron-rodgers-takes-advantage-of-free-plays-better-than-anyone/
I don't know how to give him a bump over the NFL average here but I think he deserves a bump based on what I have said about him being clutch, better than average in the first place, and the absolute very best (by a really wide margin) at generating offsides plays. The 40% number I came up really wasn't that far off. In the end, as I have mentioned, the comparison is still super marginal in the end.
Also, other professional analytics teams such as the NYT and 538.org all mention that the most basic analytic truth to all this is that the later it gets in the game, and if you are trailing on the scoreboard, the more aggressive you should be, not less.
https://www.nytimes.com/2014/09/05/upshot/4th-down-when-to-go-for-it-and-why.html
https://fivethirtyeight.com/features/nfl-coaches-are-finally-getting-more-aggressive-on-fourth-down/
https://subscribers.footballguys.com/apps/ddform.php <<<<FBG Data Dominator Tool
That 2 pointer study:
https://www.sbnation.com/nfl/2020/1/30/21112250/two-point-conversion-percentage-stats-success-rate-extra-point-nfl-dorktown