On a genealogy website I found a fair amount of the above in ancestry trees but these showed Jean Hope’s parents as John Hope 2nd Earl of Hopetoun and Jean Oliphant and she was born at Abercorn!
Never, ever, trust anything you find on line, and especially don't trust trees submitted to commercial web sites like Ancestry, MyHeritage, FindMyPast and such-like - even FamilySearch - unless it's an image of an original document, and even then there can be errors.
According to the index at Scotland's People (SP), there are three surviving records of baptisms of children to Thomas Burgess and Jean Hope, all in Kirkpatrick Fleming, between 1784 and 1792.
Have you looked at the baptism records of the Burgess/Hope children that you can find? Do they tell you what Thomas Burgess' occupation was? Was his occupation consistent with his wife being a peer's daughter?
Though it doesn't actually matter because there is evidence to disprove it.
According to SP, Lady Jean Hope, daughter of John Hope, Earl of Hopetoun and Jean Oliphant, was baptised on 18 December 1766 in the parish of Abercorn.
According to
www.thepeerage.com, she married Henry Dundas, Viscount Melville, on 2 April 1793 and then Thomas Wallace on 2 February 1814, and died in 1829. I know, don't trust what you find online .... however ....
The Reading Mercury of 1 April 1793 has the following announcement: "Married, by special licence, the Right Hon Henry Dundas, Secretary of State for the Home Department, to the Right Hon Lady Jane Hope".
An announcement in the Bristol Mirror of 19 February 1814 reads, "On Wednesday, the 16th inst, by special licence, by the Hon and Rev the Dean of Windsor, The Right Hon Thomas Wallace MP to the Right Hon Jane, Viscountess Melville".
And according to the register of burials in the parish of Haltwhistle, Northumberland, Jane, Dowager Viscountess Melville and Baroness Wallace, 33 Portman Square, St Marylebone, aged 62, was buried there on 27 June 1829.
So it appears that
www.thepeerage.com is correct about Lady Jane Hope.
If your Jean Hope died in 1830, her life clearly overlaps with that of Lady Jane Hope, therefore they are two entirely different people, therefore those online trees are plainly wrong and you can safely dismiss and ignore them.
And in future, don't bother with commercial web sites unless they have copies of original documents. And definitely don't trust them.
Don't trust me, either - take a look at those original documents to confirm that I'm not just making it all up like the lazy sloppy researchers who have submitted those trees without checking that they are correct. (I would have posted extracts but I might have fallen foul of RootsChat's copyright rules.)
See also
https://www.rootschat.com/forum/index.php?topic=714261.0