This suggestion bears similarity with the already existent and widely criticized eliminative materialism, although it is reminiscent of reductionist accounts of mental phenomena as a whole. Indeed, your recognition of the connection between the Humean temptation to reduce all mental phenomena to the impressions themselves (at the cost of an existent subject that actually experiences the said impressions) and the notion of reducing all such things as conceptions, judgments and ideas to certain facts is a natural event, for there just is such a connection between the two methods.
Granted, in one the 'self' and its functions are being reduced to a very messy ontological group of entities (that Hume still considered such impressions to be 'mental' despite ridding of the 'self' welcomed Kant's swift criticism and objection that an active agent is required for all the facts of experience to be accounted for) and in the other the 'self' is being reduced to a precise ontological group of entities, such as particular neural patterns in the mind. But there is occurring in each an austere reduction of certain supposed facts regarding the unseen self to supposed facts regarding seen particulars. In each case the notion of the self, be it a substance or not, is being equated to that which is a part of our experience. And in each case our judgement is being reduced to the parts that constitute it.
But this view has very hard-to-swallow consequences, that many philosophers have deemed crippling. For one, it is very questionable if everything the eliminativist thinks can be reduced/equated to some other ontological entity can indeed be so. For one, it seems that the old notion that the whole is greater than the parts still sticks. So for example, the judgement, which is obviously a mental act, in analytic philosophy has been found to be in no way reducible to its parts, or 'terms'. Indeed it is just the other way around. This has even been known since Jean Buridan. For example, take the classic analytical example of the 'Morning Star' and the 'Evening Star'. Each term in itself has a different meaning, but each term nonetheless names the same thing. The only way in which one can know what the single term refers to is in the whole of the judgement as represented in the proposition.
Now there is of course disagreement about what exactly such things as 'proposition' and 'term' in logic signify in the mind, but what is clear as day is that they signify something in the mind. This is why there is ambiguity in our referencing after all; different terms in themselves signify different concepts in our minds, so that it is only in a judgement that such terms can be accurate and (using medieval jargon) supposited as referencing certain things. There is no problem in the terms themselves; the evening star and the morning star 'are what they are'. The problem is a result of our attitude, our differing concepts, which are obviously the product and act of the collective 'mind'.
Furthermore, it is clear that the judgement, the proposition is in no way of the world in the same way the terms, or possible objects of referent are. We cannot experience 'The man is sad'. We can experience 'man' and 'sad', but we cannot experience what is represented by the conjunction of both, nor the truth hidden within the proposition. That which the proposition represents most immediately is called the 'enuntiable' and it is classified in medieval logic (which is largely neglected by contemporary thought, though it is finding a strong comeback in the likes of D.H Armstrong and other moderate realists), along with our concepts as entia rationis, or beings of reason, products of the mind, and not of the world.
Now as to what this 'mind' is is another matter entirely. Wittgenstein held the 'self' to be the unseen limit upon the world. But what is apparent to many of the analytic tradition is that such a self, such a limit, is not of the world. It is not identifiable with anything that is experienced in the world, precisely because of its nature, and not (contra Hume's fork) because it lacks meaning.
As to Quine, I'm not sure it is accurate to inflate his points about the synthetic/analytic divide to have precedence over discussion about the self and its concepts. For we are not speaking about knowledge and meaning we possess nor the ways in which we can represent such knowledge and meaning (which was the main linguistic concern of Quine). We are rather speaking about conditions for logical reference, which even Quine and his ontological relativity would be hard pressed to find irrelevant to logic itself.