The purpose of protected questions is, according to the message presented on such a question, "to prevent 'thanks!', 'me too!', or spam answers by new users." The more verbose "Protected questions" privilege page"Protected questions" privilege page agrees on the purpose. And the FAQ question says this is why protected questions exist:
Some questions are protected because they are expected to attract either spam or users -- often new users -- who may mistake the site as a traditional forum, posting "noisy" answers such as "Thank you" or "This worked for me" or "I'm also having this problem".
Considering this, shouldn't a user be able to answer protected questions if they have gained at least 10 rep on any stack exchange site? If they've already demonstrated one site that they are familiar enough with the SE Q/A format to not be a drive-by useless answerer, shouldn't that confidence in them carry over to other SE sites?
Or maybe at some level of combined SE rep this restriction should be waived. I find it silly that at 15k+ combined network rep, I'm unable to answer a protected question on a SE site I've just started using.