Ethernet via wifi internet sharing not working in Sierra

I have used wifi sharing (internet coming in via ethernet) for years and years on my old iMac (circa late 2006) and it was one of the things I liked best about it. Simple and elegant. I never had any problem with any other device connecting through the wifi sharing. However, I just bought a new iMac, running Sierra with all available updates, and internet sharing (from ethernet using wifi) simply does not work.

All sharing boxes are ticked correctly. The light is green. The desktop shows the gray wifi arrow pointing upward, indicating that it is sending out SOME kind of signal. My devices can see my computer/network. But when I try to connect them, they show "no internet connection" for my computer/network. But I DO have perfectly good connection coming in to my iMac on ethernet, as evidenced by the fact that I am here on the internet now typing this question.

There seems to be some sort of technical glitch or disconnect with what the iMac is putting out through the wifi signal. I have seen numerous other complaints about this, most without answers.

I have already spent close to two hours (total) on the phone with two different AppleCare people, and now they are seeking help from "the engineers." All I can say is that this appears to have been an issue for at least six months, based on my online searches for suggestions; so if the engineers pretend that it is something new, shame on Apple.

Has anyone ever gotten this corrected? I do not want to be chained to my desk nor have to buy a router!