Hi Peter, there are many old Briggs engines out there with worn-out rings, it is probably the most common wear problem. You'd think the aluminium bore would be a bigger problem, but it has a much bigger surface area than the rings do. I suggest you measure the bore diameter if possible, just 1 cm down from the top, to see if it is worn before spending money on new rings.
There are some especially critical points regarding re-installing the piston and rings. These are not the only things to be careful about, as you will have noted from the video you've seen, but they are extremely easy to mess up. First, do not take the piston off the connecting rod, or you will have problems putting it back on the right way around. Second, put the piston in the right way around: the gudgeon pin is offset, it is not on the centerline of the piston, and if you put it in backwards you will wear one side of the bore excessively. Third, put the connecting rod cap on the right way around. It was bored at the factory with the cap installed, and if you put the cap on backwards the two halves of the big end bearing will not match (there will be an offset in the big end bore), which will cause early failure. Fourth, oil the crankpin before you assemble the big end bearing. Fifth, clean the bottom of the rod and the top of the bearing cap and ensure there is no dirt in the joint between them, then assemble them by hand, don't pull them together by doing up the bolts. Then tighten the big end bolts to the right torque, don't just "do them up". On such a low speed, low stress engine, just doing them up is probably fine after you've used a torque wrench enough times to be accustomed to what the right torque feels like, but with a small bolt in an aluminium rod, until you get the feel of it, it is very easy to get it wrong in one direction or the other. Remember, tighten the two bolts in stages, just snugging both of them first, then bringing them up to torque alternately, a couple of Nm at a time.
There are fifty other things you have to get right, I've just focused on the ones people are prone to getting wrong.