Can someone extend their visa-exempt stay to apply for a new Non-B visa later?

I am pretty sure that is on the individual, not the company, for not getting the re-entry permit at the airport before leaving. Nothing to do with the company. Also, as someone else mentioned, you get the Non-B via the embassy. It is valid for 90 days. Once in country, you use the Non-B at immigration to get a one year "extension of stay" based on having entered on a Non-B. However, you dont have a Non-B anymore, because it has expired. You have an Extension of Stay based on the Non-B. You will need to check to see if the canceled your work permit. If so, you are not just starting over with a Non-B, you would be starting over with getting the work permit as well. I dont think the extension would impact an Non-B application. An extension is issued by the Dept of Immigration while the Non-B will be issued by the Ministry of Foreign Affairs.

"Someone's Non-B was cancelled on arrival because the company missed getting the re-entry permit before they travelled" - no, the person who travelled did not get a re-entry permit

The work permit is not still valid. You can’t be on visa exempt and have a valid work permit. It became invalid as soon as they left without a re-entry permit.

Many companies want people to apply for the Non-B visa outside of Thailand as the paperwork required is considerably less. If the company would provide the extensive paperwork required to apply inside Thailand he could. But why would they want to extend, they aren't allowed to work on a visa exempt entry? Although extending it won't affect a visa application. They will likely also require a new work permit to go with the new visa

Are you sure you are on an B visa and not an Extension of stay for employment, as I always understood the EOS is cancelled your work permit is gone and you have to start all over

Do you do 90 day reports ❓.. If yes you had an EOS not a B visa

And until you sort this out you can't work either