Some years ago, after work I went to my golf club to play a competition. I teed off and hit my second onto the green. I then looked in my bag and to my horror I realised I had left my putter in the car; a couple of days earlier I had been on the practice putting green and when finished I just put the putter back in the car and not the golf bag. Anyway, not knowing whether I could go back to get it, which I wouldn't have done anyway as I would have held up the other players in my group, I decided to putt with my driver. It was a 9-hole comp and I scored 19 stableford points, equivalent to 1 under handicap (I was off 8 so played to 3 I think).
Although it is a long club I much preferred to putt with the driver in those days rather than another club simply because it had the least loft (I think I used to use an 8 degree Callaway VFT driver in those days) and it was easier to 'sweep' the ball, giving me more feel. The problem with using an iron was that the ball would jump in the air and you would get more backspin, therefore making the shot more inconsistent. And if you try to 'blade' an iron you have to be pretty accurate, again making consistency difficult. These days I would probably not use a driver as the heads are too big and so more cumbersome; I'd probably use a 1-iron instead (yes I still carry a 1-iron as it's one of my favourite clubs for driving on short par 4's and long fairway shots). And I still don't carry, or even own, any hybrids.
I have seen a number of professional players use a wedge on greens, some to putt, but some to actually chip. I remember Paul Azinger I think breaking his putter over his knee one time (of many times probably) and used an iron to finish his round. At the Riviera Country Club in California, the 6th hole has a bunker in the middle of a wide green...
... and I have seen players (professionals) chipping over it if they're on the wrong side to the flag. Not sure what year it was, but I've also seen John Daly chipping on a green at St Andrews because of the undulations between him and the hole, and I think he might have even holed his shot.
So, it doesn't really matter what anyone thinks the rule should be, the fact is any club can be used anywhere, even a putter off a tee, which is not as stupid as it sounds - the putter has the least loft and therefore can keep the ball low and will run (useful into wind). Only problem is the shaft is very short and so distance will be lost. And of course the depth of the clubface is so small that hitting the sweet spot becomes an issue.