I would say that Lallans and Doric are closely related (dialects even) and can be generically called "Scots". Because of Scotland's population distribution, Lallans is more an urban language while Doric is more rural. Indeed, Doric takes its name from the Greek for rustic. Doric has probably been less influenced by modern English than Lallans has simply because the rural North Eastern communities have been more isolated than the urban areas of the Central Belt.
Burns was an Eighteenth Century ploughman whose father came from North East Scotland. His language is therefore rustic and he would certainly have had a good understanding of Doric from his father and uncle who both moved from the Mearns to Ayrshire in search of better farms. I personally feel that Doric speakers have less difficulty reading Burns than do speakers of Lallans. Not only is Doric less "modernised" than Lallans and therefore more in tune with Burns' 18th C. language, but the concerns of country folk today are often the same as those of Burns' day so Doric speakers are probably more empathetic to Burns than are the more urban Lallans speakers.
For the past 300 years there have been policies of discrimination against Scots (and Gaelic) in the Scottish education system. All lessons are conducted in English, Scots languages are not taught and, until recently, pupils could be punished for speaking Scots within the school. In such a climate it is hardly surprising that Scots is rarely written. There have been valiant efforts to formalise spellings and codify grammar, but only in some obscure university departments. The public at large never learn how to write their own language correctly so how can they be criticised for not doing so?