Thanks for that info, but unfortunately none of that relates to my issue. Even if I haven't set up a Screen Time passcode there should be an option in the same place to turn one on. On my son's phone there was simply nothing after the four main Screen Time control areas. By chance I figured out the problem.
It was apparently set up under family sharing as an parental-controlled account. because of that my wife and I can go on our phones, see our son's AppleID, tap that and THEN we go into Screen Time settings for him - including the option to change the passcode. On his device it only gives the option to change the four main settings, on our devices we have all of the options.
So that was the solution to our problem.
More info for others who might find this: Having it set up that way seems to make the Screen Time settings travel with the AppleID - no matter the device. That's good, and you can even setup the Screen Time settings to apply in total across all devices the ID is signed into, instead of device-by-device.
The annoying thing is that I could find nothing online that explained that. I tried looking to see if ST was device independent, or could be associated with an AppleID (like his apparently is), but couldn't find that simple explanation anywhere. Not saying it doesn't exist, but one would think that would be come up pretty quickly in an online search.