IMO, there are lots of misconceptions in this thread:
1. Most flats are caused by compressing the tire down to the rim and pinch flatting the tube. Punctures do cause some flats but it's less common. Pinch flats are much less common with high volume tires, thicker carcassed tires like BMX or MTB. It's really all about volume, for example I've used Compass tires that have paper thin sidewalls without flatting for a while now despite using them on fairly rough gravel.
2. I started racing USCF in 1986. Back then, everyone trained with a frame pump and several tubes. I'm not sure what the OP is talking about. Flats happened all the time.
3. Tubeless technology is really good and didn't exist a few years ago. I've personally never flatted any tubeless setup and I have it on three bikes ranging from 23mm road to a fat bike.
4. Tires are sooo much better in every way these days. Even cheap tires ride much better and are much less flat prone than the Avocets I used to race on. Tires are much more confidence inspiring.