Is David's first wife's forename known, and did her letter to Cornwallis have a townland (address)? I looked in the 1831 census for Magherafelt parish. There were 3 Burns households. All lived in Coolshinny and all were Church of Ireland (so protestant). Coolshinny is on the outskirts of Magherafelt town. Magherafelt Church of Ireland baptism, marriage & burial records start around 1720. You could search them looking for a marriage and the 6 childrens baptisms. Also for his wife's burial. There's a copy in PRONI (the public record office) in Belfast.
As far as the United Irishmen are concerned, the official records are split between Dublin and Belfast. PRONI in Belfast have quite a lot including lists of prisoners and suspects. Also letters between Castlereagh and General Nugent and Pitt the Younger (Prime Minister) about what to do with the prisoners. A useful source in PRONI is the Black Book of the Rebellion MIC 575/1. From what I have read in those papers, the vast majority of the rebels were released without further action taken against them. A few leaders were executed. The only others against whom action was taken were middle ranking "officers". It was feared that if they were allowed to stay in Ireland they might foment further rebellion. (The French Revolution had only been 6 years before and the Government was very nervous that the same could happen in Ireland). So the middle ranking trouble makers got sent to Australia or in some cases were allowed to make their own way to North America, with their families, on condition of not returning. So possibly his transportation indicates David's status within the United Irishmen. As you say martial law was in place at the time and most of the decisions about people's fates were fairly arbitrary and carried no right of appeal. They were seen as terrorists and had to be dealt with as such.