I never said patch, and in theory, you can't have a puncture without some belt weakening/damage. The surrounding strands are just strong enough to support it. Plugs are the only type you can install without breaking down the tire. The only approved way of repairing a tire is a plug/patch combo. The one piece patch has a tapered plug attached to it which is pulled through from the inside of the tire. The plug fills the hole so dirt and water can't work the patch loose over time like a plain internal patch could. It also means that with centrifugal force, it cannot come out at high loads and speed.
When plugs were allowed, I had numerous over the years with no problems. I am not saying that they can't work, just that they have a greater potential to fail. I also had no problems with the old internal patches when installed properly. The new industry standard is the plug & patch, is the only type a tire shop is allowed to install. They still make plugs, and sell them for temporary use, or for use on a vehicle that is never driven on public roads (off-road use only).
Bottom line is that yes, for $5 you can buy a DIY plug kit, and I always have one with me. But the reason you pay $25+ at a shop is because they do a proper repair (or at least they should!).