The bottom line is what you want to do/see. On the one hand you always go during March madness, so I asssume your primary purpose is to bet on basketball games. That being the case, you can sit in one of the megabooks on the strip and not need a car. Of course you will pay exhorbitant prices for food, drinks, lodging, etc, (unless you are comped for all those). You also said that you always complain that you don't see enough of the strip!? I don't think a car will make much difference in that regard. I'm not sure what you mean by overcrowded parking lots, since I've never had a parking problem either self park or valet at any of the strip hotels, (maybe once or twice at Harrah's). Quite honestly, I don't think March madness is much worse than any other time of the year. On the other hand taxi and luggage fees with at least four of you will probably be nearly as much as the car rental - we always use Hotwire or Priceline and get fees approx 50% less than the other travel sites (expedia, Orbitz, travelocity).
Try staying downtown and commuting to a big sports book on the strip - you can walk from your room to your car in less than 2 minutes at most of the downtown properties, and the gambling choices are just as diverse as on the strip. We've been coming to LV a couple of times a year on average for at least 25 years and have never gone without renting a car. I can't imagine doing without one. Our stays usually average about a week in duration, and we generally only spend one day on the strip - just to walk through what's new and maybe see a show. There's way too much to see elsewhere in LV that is not accessible without a car.
You also mentioned a trip to Boulder Highway - there are plenty of good sportsbooks there with lots of parking and cheap hotel rooms. Then you could just use the car to get to places you really want to go. Sam's and Sunset station both have decent places to watch and gamble on the games. It sort of sounds like you guys need to plan a trip to LV at a different time of the year so you wouldn't have conflicting interests!