There's already a mechanism that gives privileges to "high" rep users of other sites. It's called the association bonus:
If you are an experienced Stack Exchange network user with 200 or more reputation on at least one site, you will receive a starting +100 reputation bonus to get you past basic new user restrictions. This will happen automatically on all current Stack Exchange sites where you have an account, and on any other Stack Exchange sites at the time you log in.
This bonus unlocks nearly half of all the privileges (notably upvoting, commenting, and flagging).
I think it unlocks only so much for a reason. Close voting in particular makes sense as a privilege. If a question needs closure, a user with 101 reputation can flag it, pushing the question into the queue where other users can decide if it should be closed. If the question shouldn't be closed, the flag gets declined, and enough of that gets you flag banned.
Consider this: I got the ability to close vote on one site through nothing but a single answer (because the site is in beta and the question I answered was in the HNQ). As a new user, I'm unfamiliar with the meta discussions where the site decided what's on and off topic. There's nothing to stop me from voting to close the wrong questions, with little to no feedback that I'm doing it wrong. My experience on other sites doesn't change any of this. That's why I don't vote to close on that site at all.
Also, people have discussed making it so that the bonus doesn't count towards upvoting, although nothing has been decided or done yet.