A picture can be worth far more than a thousand words and an ounce of knowledge is worth a pound of guesses.

Sheldon Brown RIP created one of the best sources of information on cassettes on the web. It has lots of pictures with easy to understand explanations plus work arounds. I used this when I put together my first cassette hub back in 2007. I put a 7 speed cassette on a 6 speed freehub by changing the spacers. (yes, I'm a late adapter and a retrogrouch)

https://www.sheldonbrown.com/k7.html

One note, I call the UG (Uniglide) sprockets "twist tooth". When Shimano introduced them they offered an improvement in shifting of previous tooth designs but HG (Hyperglide) sprockets shift so much better.

One problem that I've run across with the twist tooth UG sprockets is that on a hard shift with a worn out upper derailleur pulley I've had chains bend to the angle of the twisted teeth.

Most modern replacement chains for older bikes are designed to work on 6-7-8 speed freewheels and cassettes. 9 tooth and up cassettes definitely need narrower chains. That's why chains for 9, 10 and 11 cassettes are available.

When Shimano introduced their SIS indexing system in the mid 80's they started making upper pulleys with some side to side lateral movement. This allows the chain to self center on the sprocket which is key to effective index shifting. Worn out upper pulleys can be rocked back and forth at an angle as well as side to side. They need top be replaced.

BTW, RDs made for indexing work GREAT with friction shifters. The side to side float of the upper pulley pretty much eliminates trimming the lever after shifting.

verktyg