The primary function of the conical ends of the lug nuts it to center the wheel. If the lug nuts are loose, you'll lose that function, which can unbalance the wheel and indeed contribute to your bearing problem. Standard nuts don't provide any centering whatsoever. I make it a habit of checking the lug-nuts a day or so after changing a wheel. Loose wheels are a significant safety risk because they can eventually come off. A wheel coming off can cause you to lose control of the vehicle. And since a golf kart has none of the safety features of a modern car, losing control in a car might even be safer even though it usually moves at much higher speeds.

Due to the thick washers, the nuts that you used didn't fit completely on the studs. Lug nuts are usually higher than standard nuts to provide more bearing surface on the threads. Your modification puts a lot more stress on the thread of the studs. If you strip the tread on the studs, you're looking at a more expensive repair than a couple of lug nuts!

One of the secondary principles of good engineering is parts count reduction. That is, e.g, not using more different types of fasteners than necessary. This makes assembly and inventory control easier. So if a non-standard fastener is used somewhere, there is bound to be a reason for that. It is seldom a good idea to change those things, unless you know the design context.

It didn't seem very risky when I drove the golf cart until the wheel fell off (causing the problem Mark was trying to remedy.) It just ground to a halt....

(No, I really don't recommend following my lead, but I thought I'd add the data point.)