I don't think a philosophy site is the best place to explain the psychology of philosophers' practical stance(s) towards morality. But some response can be made.
Fact/ value distinction
If moral realism is right, then values and valuations are facts or factual, so that there is no logical bridge-crossing from [factual] 'is' to [factual] 'ought'. For whatever reason, a large number of philosophers subscribe to some version of moral realism. In which case there is no is/ ought gap or problem for them.
Reading (below) : DeLapp, Putnam.
Hume assumes a fact/ value distinction and responds that while logically we cannot derive an 'ought' from an 'is' (if there are no normative premises there can be no validly deduced normative conclusion), this is besides the point morally.
Hume observes that 'Morality ... is more properly felt than judg'd of' (A Treatise of Human Nature, III.1.2). We can continue to use the term 'moral judgement' as long as we realise that a moral judgement does not refer, despite appearances, to 'real' moral properties in a mind-independent world. Rather, a moral judgement expresses emotions (or 'sentiments'). Not just any emotions one happens to feel, however. Very far from it. It must express a moral emotion. And for a emotion to count as a moral emotion, there are two prerequisites or conditions.
Moral judgement - condition 1
Hume stresses that :
it is often necessary, we find, that much reasoning should precede, that nice distinctions be made, just conclusions drawn, distant comparisons formed, complicated relations examined, and general facts fixed and ascertained. (Hume, An Enquiry concerning the Principles of Morals, I.)
Briefly put, moral judgement requires that we ascertain the empirical facts about a situation, distinguish them, see what they appear to imply, make sure that we are reflecting on the precise situation at hand and not on one merely deceptively similar to it, draw on relevant historical parallels : and so on. We need an accurate factual fix on the situation. Moral judgement is no flip emotional reaction; it needs to be factually informed.
But does Hume offer no criteria for just how much information, of what exact reliability, we need ? Partly, I think, he assumes that in many case we can use sociocultural standards to determine the degree of rigour and the extent of evidence required in making a moral judgement. Importantly, he doesn't leave us merely floating in this boat, however. In the Treatise of Human Nature, II.1.3 he lays down rules for the scrutiny of evidence, probability, and belief.
Moral judgement - condition 2
That is one condition for moral judgement. The other is that we must take a third party point of view. Of the moral agent, who judges, Hume says :
He must here, therefore, depart from his private and particular situation, and must choose a point of view, common to him with others. (Hume, An Enquiry concerning the Principles of Morals, IX.1)
When these two conditions are met, we experience 'certain peculiar sentiments of pain and pleasure' - of approbation and disapprobation. Briefly, moral judgements express our sentiments - our emotional reactions - of approbation and disapprobation under these conditions. Moral judgement is causal through emotion, not cognitive through reason.
The role of sympathy in human nature and moral judgement
The whole picture rests on a picture of human nature as containing a capacity for sympathy which enables us (in Hume's language) to depart from our private and particular situations and take the standpoint of what Adam Smith would later call 'the impartial spectator' (Adam Smith, The Theory of Moral Sentiments, I.5). In this detached, third-person perspective, and factually informed, we experience the moral emotions which we express in moral judgements.
In a distinction which Hume draws between the natural and the artificial virtues, the arousal of sympathy occurs less directly in the case of the artificial virtues than in that of the natural. But sympathy is still the key to both. (Hume, A Treatise of Human Nature, III.2.1.)
In view of all this, there is no logical bridge-crossing in Hume's ethical theory any more than there is, on very different grounds, for the moral realist.
Reading (below) : Hume online references, Cohon, Harrison, Mackie, Adam Smith online reference, Thomas.
Theory and practice
If ethics is a theoretical discipline, a philosopher can quite properly scrutinise it philosophically without any practical requirement to act on it. I could study political philosophy without any political involvements and without seeing myself as having any practical commitment to engage in political action. The same holds for ethics. One may regard it as an outmoded way of thinking, and the moral life a mere curiosity, yet still do deep and intricate conceptual and logical work on both. Work on which one closes the door when one leaves one's study and walks away without walking the walk.
Reading (below) : Aristotle, Nicomachean Ethics, Bk X, where contemplation (theoria), detached from practice, is given priority over moral action. (This interpretation of Aristotle is defensible but open to dispute but it has certainly informed a stream of philosophical thinking which denies practical relevance to philosophy. Aristotle did not go so far as that.)
Amoralism and rational choice
This leaves only your philosophers who 'don't really feel it is right nor necessary to behave morally, but do so because there are subjective advantages'. May they not be rational choosers (subscribers to Rational Choice Theory) who act morally, at least so far as concerns behaviour and without any sense of duty or obligation, because in any of a number of ways it 'pays' ? Their position is foreshadowed by Glaucon's account of morality (though not his own position) in Plato, Republic, Bk II.
Reading (below) : Plato, Republic : https://www.gutenberg.org/files/1497/1497-h/1497-h.htm & J. Baron, Morality and Rational Choice, SBN 10: 0792322762 / ISBN 13: 9780792322764. Published by Dordrecht, Kluwer Academic Publ., 1993
Plato, Republic : https://www.gutenberg.org/files/1497/1497-h/1497-h.htm.
Aristotle, Nicomachean Ethics : https://ebooks.adelaide.edu.au/a/aristotle/nicomachean/
David Hume, A Treatise of Human Nature, 1739-40 : https://www.gutenberg.org/files/4705/4705-h/4705-h.htm
David Hume, An Enquiry concerning the Principles of Morals, 1751/ 1777 ed. : https://www.gutenberg.org/files/4320/4320-h/4320-h.htm.
Adam Smith, The Theory of Moral Sentiments, 1759 : https://www.earlymoderntexts.com/assets/pdfs/smith1759.pdf.
J. Baron, Morality and Rational Choice, SBN 10: 0792322762 / ISBN 13: 9780792322764. Published by Dordrecht, Kluwer Academic Publ., 1993.
R. Cohon, Hume's Morality: Feeling and Fabrication, ISBN 10: 019959497X / ISBN 13: 9780199594979
Published by Oxford University Press, U.S.A., 2012.
Kevin DeLapp, Moral Realism, ISBN 10: 144116118X / ISBN 13: 9781441161185
Published by Bloomsbury Academic, 2013.
J. Harrison, Hume's Moral Epistemology, ISBN 10: 0198750374 / ISBN 13: 9780198750376
Published by Oxford University Press, 1976.
J.L. Mackie, Hume's Moral Theory, ISBN 10: 0710005253 / ISBN 13: 9780710005250
Published by Routledge, 1980.
H. Putnam, The Collapse of the Fact/Value Dichotomy and Other Essays, ISBN 10: 0674013808 / ISBN 13: 9780674013803
Published by Harvard University Press, 2004.
Geoffrey Thomas, An Introduction to Ethics, ISBN 10: 0715624318 / ISBN 13: 9780715624319
Published by Gerald Duckworth & Co Ltd, 1997 : 53-8 [Hume].