My opinion (as a member at Roger's club). Motorized carts are the stigma, if anything is. They are all over the public and resort courses (just like the guys who meet you in the parking lot at many of these places and want to carry your clubs 25 yards to the club house). However, when motorized carts are used under conditions such that walking won't work for the golfer, they're o.k., and in those cases I don't consider them to be a stigma.
It's a cultural thing we could learn from the Scots, from whom we learned golf. At Royal Troon, one of the stuffiest clubs I've visited, they may have motorized carts for the those with medical problems, as many Scottish clubs do, but I didn't see them. On the other hand, they'll rent you a bare bones pull cart for several pounds or you can use your own. One of their members loaned one of our group his personal pull cart to use so we only had to rent three from the club (although the member was not permitted by club rule to share a drink with us, as visitors, in the club bar -- go where you want with that).
Exercise. Walking a course is low level exercise, but it does have value and is better than none at all. Funny thing is that my sense is that relatively more younger golfers than older golfers ride at our club, which is not to say we don't have lots of younger members who walk (at age 67, I'm one of them). Still, I'm amazed at all the young, apparently heathy young people who ride. (I'll ignore the we-need-the-cart-to-carry-the-beer argument.) For me walking is an essential part of the game. If not mistaken I believe the PGA, LPGA and USGA buy into that, too.
Walking carrying vs. walking pushing or pulling. If only the old or infirm use push/pull carts, then why have the caddie carry your bag, young man? If you need him or her for advice, bring the caddie along but you carry the bag.
Speed of play. Except for those so-called golf courses with long walks from green to the next tee (e.g., typical residential development "golf courses" in this area), speed of play has everything to do with the players' approach to the game and virtually nothing to do with walking vs. riding. Some golfers play slow, some medium and some fast.
Revenue. Raise everyone's dues? Or charge $X per round of golf regardless of whether the golfer takes a buggy? But wait, in joining the golf club and paying dues aren't we already paying to play golf. We're a golf club. We join to play golf and pay dues to play golf. We ought to be able to walk and carry or push/pull any time we want to, at no extra charge. But, riders do pay extra for carts (easy for me to say because mostly I walk). That's what we do now at Roger's club, which is the way it ought to be.
Regarding Golfweek, also from the same issue, see:
http://www.golfweek.com/story/toy-box-031009