Can we extend Hume's skeptical conclusions to the treatise itself? How far reaching is Hume's skepticism, and if it is all-encompassing, then what is (as he sees) the purpose of his work?
Scholars have always disagreed about the nature and scope of Hume's scepticism in the Treatise. You take a particular, and interesting, angle on this by suggesting (if I have it right) that if Hume's arguments in the Treatise are sound then they undermine everything in the Treatise itself. Hume projects a scepticism that devours his own text.
This is a point well worth considering. Let's see how the case could be made. There are three sources of possible knowledge for Hume : (1) 'demonstration' or deductive logic or reasoning, (2) induction, and (3) perception or sense experience. All three are assailed in the Treatise. Deductive logic can only be defended circularly by the use of deductive reasoning itself (T I.4.1). Inductive or probabilistic reasoning has no firmer foundation than 'Custom' : T, 'Abstract' : 'We are determined by custom alone to suppose the future conformable to the past'. Perception carries no guarantee that what we receive by way of 'impressions' resembles the objects in a supposedly external world which causes them (T I.4.2).
Insofar as the Treatise employs deductive or inductive reasoning or makes appeal to the deliverances of the senses, then it seems self-defeating. It destroys the supports on which it stands.
But I do not think that this defeats the purpose of the work. Recall the title, 'A Treatise of Human Nature'. Hume is trying to lay the foundations of 'the science of man' (T, Introduction). He wants to discover and to tell us what human beings are like. We perceive; we reason deductively and inductively. Hume never suggests that we should or could stop doing any of this. What he does is to expose the false basis on which so many of our assumptions in doing these things rests. There is a parallel. It is fairly clear that Hume saw himself as engaged in the case of human beings in the same kind of fundamental explorations of the physical world that Newton had made in the 'Principia'. Newton leaves us with our familiar world of appearances - the world looks no different after we have read the 'Principia' from before - but the explanatory mechanisms Newton reveals are remote from any that the ordinary person of the time could even have imagined.
Perhaps this is especially clear in the case of something we have no yet mentioned, namely morality. In T III Hume devastates the entire traditional understanding of the foundation of morals. (a) It is not based on reason. (b) Moral truths cannot be perceived; and (c) they cannot be arrived at inductively. (In the Treatise he does not even consider divine revelation : too much 'imperfection attends our idea of the Deity' - 'Appendix'.) But Hume has no intention of undermining morality. It is a stable and intrinsic part of human nature. Only, he wants to make clear that 'Morality ... is more properly felt than judg'd of' T III.1.2). We are to seek its origins in our emotions, particularly that of sympathy.
To sum up : Hume leaves the surface of human nature exactly as everyone else finds it. People reason deductively and inductively; they perceive; they make moral judgements. Nothing is changed and Hume has no desire to change it. What he does want to do is to show that the foundations, the epistemological and metaphysical bases, of these activities are quite without the justification they are widely thought to have. It is as if we can continue to walk with assurance on a floor of which the supports are entirely different from what we suppose them to be.