The way I made my cradle (550 nylon twine) is by it touching only the j-hooks and not touching the tonering.
I wanted to make sure my banjo sounded the same whether standing or sitting (with or without the strap).

I've had cradle straps on my banjos and it doesn't seem to make any difference in sound, tone, playablity. In fact, it makes it much easier to play as you aren't always pushing up the neck to get it in the right position.

You could weave it back and forth between the hooks and I don't think it would touch the tone ring at all. In my mind, I can't imagine it mattering.

‘Tone’ is a very subjective area. It is easy to identify volume in terms of loud and soft (not loud). Having been round banjos for around 6 decades and working on them for a few less, I understand what goes into the construction of them and what affects what we hear. To my mind, it is a tone chamber (area between the inside circular rim and the vibrating head that causes the tone. Even enclosing the chamber with a resonator affects things. I realize that this is a very simplistic explanation because the materials that go into the parts that make up this area also have an effect as does how everything fits together including the make and tension on the head.
I have used cradle straps in the past and to be honest, have never noticed any difference in what my ears hear. While the wood and the metal do vibrate, I doubt if anyone could notice the difference if a leather strap bears up against the rim/ring. Just my 2c worth

I have been using cradle straps of my own design for years after having an incident with a broken piece of leather on a hook. (I caught it), but I have never noticed a change in tone. The balance offered by the cradle gives me peace of mind and also in my opinion just tying a strap on the hooks seems to put more stress on the instrument than the cradle.

I dunno about players not being able to hear a difference .... I've heard it said some people can tune their banjos without using electronic tuners!