On the whole, I wouldn’t mind giving the judges a little leeway to affect the results. At least early on.
I’ve suggested before that each judge gets one “get out of jail free card” to use, and that each contestant can only benefit from it once. When the “bottom 3” is announced in a given week, one judge may use his/her “get out of jail free card” to make a contestant safe; if that happened to be the lowest vote-getter, than the next-lowest vote-getter would be removed.
After the field is cut down to a reasonable number, the “get out of jail free card” system is no longer in effect. So, after we’re down to, say, 5 or 6 are left, there’d be no more “get out of jail free cards”, even if they haven’t all been used.
From a producer’s standpoint, this would, at the very least, given them some fudge factor as to helping them choose who makes the touring top 10. It also allows them to save a good contestant who just had a bad week. The key would be to have the system limited so that such decisions would be the exceptions, not the rule.
Your idea wouldn’t be so bad since it would only be allowed 3 times and then its done, no more interference.