Hi Richard,
I am out of my depth with naval genealogy - suggest you try back on the ?armed forces board - I have both attestation papers and pension records for one of my ancestors who was in the army (who was born in Montrose at about the same time as your Mr Burnett) - I see from the census that your man was 'superannuated' so perhaps there are separate papers for this but again, I'm no expert.
When I call up the 1851 census record on A/try for Thomas I see that A/try suggest records for him in the 'UK, Naval Medal and Award Rolls' and 'Eng and Wales, National Probate Calendar' collections - so probably worth looking into these if you have not already.
When I search on Scotland's People or LDS IGI index for 'b*rne*' born Montrose from 1760 - 1780 I see there is a 'William Burnet' born 21 Jan, baptised 24 Jan 1779 Montrose to a William Burnet and a Mary Croll/Croal. He had elder siblings Ann, Catherine, David and Margaret. As I mentioned these records are incomplete so in theory there could have been ?ten men born in Montrose of this name and at this time ... just because there is only one record for such a birth in existence does not mean it is the one you are searching for sadly. It also does not mean that these are all the siblings of this William - there may have been more. A/try suggests that there is a record for William Burnet in the National Probate Calendar too so this could be well worth looking into to establish if there might be any link between the men.
Other avenues that I would pursue if I was in your situation:
1. Contact Angus Archives - they do charge but have access to records that are not otherwise available ... and the charges are modest
http://www.angus.gov.uk/history/archives/aboutus.htm2. Parents names are recorded on most (there are annoying exceptions) Scottish death certs after 1855. If you search on SP statutory deaths for surname 'b*rne*' born 1780 plus or minus ten years in Montrose there might be some certificates worth paying for to give you a pointer towards generations further back. I'm out of credits so can't advise on any that look promising but certainly I've had luck in locating names not recorded in the OPRs this way.
3. You can also view Wills on SP ... ones that caught my eye that I'd be looking at are SC47/40/23, SC47/40/16, SC47/40/12 ... there are others too
Let us know how you get on?