Agreed, a "restart level" button would be great.
Another thought just occurred to me with your comment "you're doing well but are simply off to a really bad start on some level". I don't know how you can be "doing well" if you're "off to a bad start on some level" with the current scoring method, where if I lose one level, I lose all my score. I've always been a little miffed by the unforgiving nature of the scoring system, but until now I didn't really think to say anything about it. Now that I'm thinking though, it doesn't really make sense (to me) to throw away any great work you might have done in previous levels, just because you fail on a new one. I *do* think there should be an effect for losing a level, but killing all your points seems unfair. How about a point deduction/reduction/fractioning instead? That way if I do really good for 8 levels and get stuck on the 9th, I won't suddenly lose all my points when I lose the first time, I will have a few chances to win and maintain at least some of my points. This would of course result in overall higher points values and scores would not be comparable with previous versions, but as the current public high score holder, I'm fine with that.
Maybe this is along the lines of what you're already thinking for your "revised scoring system" mentioned earlier?