I didn't think that any of these would solve the problem, but I tried just for kicks. Nothing changed, but thanks for the suggestions.
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OK, one of the restrictions is for setting the age limit for apps (which is what I want to do, hence the reason for using the restrictions). As soon as I set this limit, the option for 'require password' automatically changes to immediately and cannot be modified. While this should relate to the installing of apps, apparently it also applies to the updating of apps. And, no, the age restriction itself has nothing to do with whether or not a password is required. I've already checked this. Any app, no matter the age rating, asks for a password when updating.
Thanks for trying, but I've gone through every setting I see and unless there is just something extremely obvious that I'm missing, that's just how it works.