It is important to look past the racist myth that black people were not considered human by many people. The founders did not simply have a moral/metaphysical view whereby it was immediately apparent that black people had no rights. They did not merely overlook the issue. Viewing the history this way is an underhanded way to justify the lack of rights for black Americans as an unfortunate relic of how people used to think rather than the deliberate decisions of the white founders to consciously disregard a segment of the population fully conscious of the contradictions it entailed. James Otis was one prominent revolutionary figure with strong anti-slavery views. Thomas Paine, who played such a large role in fomenting revolutionary sentiment, also wrote against slavery. Many freed and enslaved black people also spoke against slavery in the language of natural rights and drew attention to the contradictions. Jefferson's original draft of the Declaration of Independence included a vicious attack against the King George's maintenance of the slave trade that was decidedly anti-slavery, although Jefferson's own views were more complicated.

In short, everyone was aware that there was some potential for contradiction here and some unease. In the decade leading up to the Constitution, there were significant relaxations in slave laws across the country, even in parts of the South. There seemed to be, for some, a growing anti-slavery movement. This movement was ultimately extinguished and reversed in large part due to the economic necessity of slavery for the Southern economy as it was constituted. Many of the founders essentially bit the bullet. To be sure, many also argued that slaves had biological limitations that prevented them from being able to fully deserve natural rights. However, it is important to understand that a) these arguments coincided with counter-arguments that the founders were well aware of, and b) they were in a sense reactionary arguments against a growing feeling of the inherent contradictions of slavery.

There is no easy answer, but the idea that there was a contradiction certainly was "in the air," if not accepted universally by the founders. There was a movement to address this contradiction that grew steadily, but ultimately failed due to a desire to keep the Southern States unified with the northern ones. The attack against the slave trade was removed from the final draft of the Declaration. Ultimately, the choice to retain slavery in the face of natural rights was just that for the most part: a choice. It was not an unseen contradiction outside of the moral outlook of the time. Ignoring that so many people did oppose this contradiction and that there was a growing legal movement against it that was snuffed out by the the end of 18th century is a nationalistic history designed to glorify the Constitution and it's founders and make it seem as though the demise of slavery came about as people's morals naturally evolved to see it as wrong. Many already knew it was wrong; the founders were well-aware of these arguments from black Americans and prominent white ones, and America was nonetheless founded on this contradiction.

To speak to a bit of your question that /u/magalouge did not get into, yes, both opponents and supporters of the American Revolution noted the hypocrisy. English writer and political Tory Samuel Johnson famously asked "How is it that we hear the loudest yelps for liberty among the drivers of negroes?" in a pamphlet critiquing American arguments for revolt and independence. Lord Dunmore, governor of Virginia, was the first British official to offer freedom from slavery in exchange for military service, building off of a threat in 1775 to free and arm slaves to keep unruly American "Patriots" in line. Offering freedom in exchange for service became official British Army policy with the Philipsburg Declaration of 1779. The Americans grudgingly followed suit years later, after years of debate and reluctance to accept black soldiers. Washington was one of the many Southerners who opposed creating an integrated army, and changed his mind only when the Continental Army's manpower shortage became dire and America risked losing political face to Britain.

American abolitionists used the philosophy of the Revolution to push for an end to slavery: Pennsylvania abolished slavery in 1780, and every state in New England save Rhode Island ended the institution in 1783 (Rhode Island followed the year after). Hamilton and Franklin both joined or formed anti-slavery groups in the early years of the Republic. Unlike the later 19th century abolition movement, there were a few prominent people in the South that spoke out against slavery. This is one of the reasons that Henry John Lauren's death in a useless late-war action is so tragic - he was the son of a plantation owner, but wrote strong critiques of slavery and could have gone on to an influential political career.

Of course, the most direct critique of the Founding Father's hypocrisy came from the thousands of enslaved people who fled to and fought for the British rather than remain with their American masters and hope for a moral epiphany that would cause them to end slavery. These men and women endured every species of privation and danger to support the royal cause. While the British occasionally failed or cruelly betrayed these people - turning them away from their camps when supplies got to thin, or re-enslaving them and shipping them to the Caribbean - black people overwhelmingly saw the British as a more reliable path to freedom.

Sources:

Shimon Schama, Rough Crossings

Gregory Urwin, "When Freedom Wore a Red Coat"