Sounds a right load of cobblers to me. Without checking out any of the lesser known versions, I can only comment that the well known Coppers version has nothing whatever to tie it in to highway robbery.

There is a song about Spence Broughton in Frank Kidson's Traditional Tunes, but it has no connection at all with STR.

My guess is that the Coppers' Spencer was either a labourer of some kind who'd fallen on hard times in one of the nineteenth century depressions, and been unable to find work. So he took to the road hoping to find work that way. When it didn't pan out, he simply returned to his family to starve there instead of on the road.

Alternatively, he might have been a gentleman who'd fallen on hard times through gambling, drinking etc, and took to the road to get away from the wife and debts.

Whatever, the song ends happily, with no talk of highwaymen, gibbets or hanging.

It sounds to me that, as happens a lot with these local histories, the author has heard the suggestion that Broughton and Spencer were one and the same, and included it in his book without checking the facts.

BTW., according to Wikipaedia, Broughton's body was hung in the gibbet for 36 years. So, while he certainly wouldn't have been the "last man in England to be punished in this gruesome fashion", his may have been the last gibetting to be taken down.