Walkers are different people than cart ballers. I’m sure the guy who walks to a meeting or takes the stairs is quicker to get to the task at hand. Walking is not faster, walkers are.
A generalization with many, many exceptions. But these anti-cart notions that riding is not faster than walking and that carts offer no net economic benefits to a club have been promulgated as facts quite regularly on this site. They are demonstrably false.
On the first, my group usually has the first time during the weekday mornings. We have a number of riders with later tee times who we let go first because there is no way to keep up with them- we routinely play a 4-ball walking in between 3-3:10, and 3-balls under 3:00. We have riding 4-balls who can play in 2:30 easily, except that they run into the maintenance crews and are forced to slow down.
Re: the economics of riding, we may think that we're smarter than the folks at Club Corp, but the company does not stay in business by losing money on such things. I am unaware of a Club Corp property that does not have a very high riding to walking ratio. My own analysis when I was looking at such things indicated that the cart fleet had the highest % contribution to cash flow. Depending on volume, $300-$400k annually was not unusual.
Re: non-riding carts, my home club and others I am aware of do allow their use, but only if rented from the club at $8-$10 per round (we have been allowed to use our own temporarily due to C-19). There seems to be more leniency toward these over the years, but other than munis, I don't think that very many privates allow the use of personal carts.
Any thoughts on the impact on the greens if these carts could be pushed/pulled through the greens? There is an impact on the pace of play when they have to be rolled around large green complexes and creeks.