Of the few criticisms of his game, does Rodgers not gamble enough when down late?

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Interceptions are often (even largely) a product of completely rational risk-taking by desperate quarterbacks. A logical implication of this is that if a quarterback is too conservative, he can throw too few interceptions, which can be just as bad as throwing too many.

Despite his various successes, it’s possible Rodgers fits this description of an overly conservative quarterback. For example, with his team down by two or more scores (9+ points) he has thrown only three interceptions out of 354 passes attempted (0.8 percent) in his career. This is typically when quarterbacks throw the most INTs, because they’re trying to get their teams back into the game, and high-risk strategies often give them the best chance to win. Overall, quarterbacks throw interceptions about 3.5 percent of the time on average in those situations, with even most great quarterbacks breaking 3.0 percent. Peyton Manning, for example, has averaged 3.1 percent, Drew Brees has averaged 3.3 percent, and even Tom Brady has thrown 2.3 percent (slightly above his career average).
Throwing too few interceptions in the first paragraph should be qualified. There is probably no doubt throwing fewer interceptions on even an average team is beneficial. But there are always specific game situations where the risk is worth it.