I stumbled on the link Forfarian posted earlier in my searches & discarded it as I wasn't too sure this was the same place, Cambuslang (Glasgow) v Kilbarchan (Renfrew) & still not convinced.
You're right. I searched for 'Burntshields' and 'Cambuslang' and didn't notice that it was in Kilbarchan - you need to scroll down quite a bit to see that. My apologies.
So no, this is a red herring, not the place where the Purse family lived.
I have looked in the LDS CD-ROM version of the 1881 census, and there is no Burntshields or Bruntshields listed in Cambuslang in that census. There is a listing transcribed as 'Hamilton Rd Lightburn', and most of the houses in the same Enumeration District are also on Hamilton Road.
Looking at the first edition of the Ordnance Survey six-inch map
https://maps.nls.uk/geo/explore/#zoom=16&lat=55.8103&lon=-4.1395&layers=5&b=1Lightburn is a row of houses on either side of Hamilton Road. Compare it with the first edition of the 25-inch which was surveyed in 1896
https://maps.nls.uk/geo/explore/#zoom=16&lat=55.8103&lon=-4.1395&layers=168&b=1and you can see that there have been big changes. A colliery has been developed at Flemington complete with rows of cottages, the side road that diverged from Hamilton Road in the older map has now become the major road, and a linking road with houses on one side has been built.
Compare this again with a modern map, and you can see that what was Hamilton Road has been renamed Lightburn Road from its junction at Halfway.
I speculate that Burntshields Cottage was one of the houses or cottages in Lightburn fronting what was originally Hamilton Road, but was renamed Lightburn Road at a later date. These may not necessarily be the same houses that were there on the oldest map.
A rateable value of £20 in 1905 would have been a fairly substantial house, rather than a miner's cottage.
From the satellite view it looks as if most if not all of the 19th century houses in Lightburn have been demolished and their sites re-built on.
Some of the more recent Ordnance Survey large-scale maps do show the names of some individual houses. If you can find one of these from about the 1920s (which are not, as far as I know, available online, but I'd be happy to be told I am wrong about that) you may be able to see which house was Bruntshields Cottage.
