Understand your frustration!
Regarding Kidwelly, there’s a history of Kidwelly website which mentions that there was a Rogers family driving the tin mining industry there in the same period. However they sound like a local family from the county town of Carmarthen who share a relatively common surname.
http://www.kidwellyhistory.co.uk/Industry/TinWorks.htmI skimmed some of the articles but couldn’t spot any professional or entrepreneurial reason Joseph and Harmer would have been attracted to move to the area around 1776. Just a correction - his son Harmer (aka John?) was about 23 and presumably already working in the medical profession, so they may have been in partnership.
It’s a good idea to try to ‘disprove’ the Elizabeth Rogers 1777 Laugharne death hypothesis.
Agree with you about the Norfolk deaths. She could have been visiting a family member somewhere outside the county (her sister was in Suffolk, for example). Unfortunately there are gaps in the newspaper archives over the period - in particular some of the Norfolk Gazette. Even if Elizabeth’s death was in the paper, the announcement may not have survived.