This seems utterly bizarre: if I have rights to edit posts unilaterally without approval, shouldn't I still have rights to approve low-rep users' edits unilaterally as well?
Yes.
And you do. Just click "Improve" when reviewing the edit, make some minor change (or lots of major changes, if the edit you're approving didn't go far enough), and click Save Edits. The pending edit will be approved instantly, and your edit submitted immediately after.
Of course, your name is on the post now, but that shouldn't matter if the edit was a good one.
Reviewers are lamentably unaccountable for their actions in the current system; someone making a bad edit - either directly, or via a suggestion - gets their name in the revision history and below the post itself, while the folks who reviewed the suggestion remain several clicks away. I kinda suspect this has led to more than a little sloppiness, hence the change.