Your excellent question is as old as the invention of calculus. As you correctly point out, velocity makes no sense if all you know is what is happening at just that instant of time. Physicists and mathematicians take the limit of the average velocity as the very definition of instantaneous velocity.
That turns out to be a very good definition, since it leads to physics that accurately describes the behavior of the world and mathematics that's consistent and interesting and useful. So people no longer worry about the question in the form in which you've asked it.
Edits to respond to comments. Edited again (as @Polygnome suggests) to incorporate the sense of the comments as well
@pjs36 Yes indeed thanks. The question really does go back to Zeno's paradox of the arrow. On that wikipedia page you can read
Zeno states that for motion to occur, an object must change the
position which it occupies. He gives an example of an arrow in flight.
He states that in any one (duration-less) instant of time, the arrow
is neither moving to where it is, nor to where it is not. It
cannot move to where it is not, because no time elapses for it to move
there; it cannot move to where it is, because it is already there. In
other words, at every instant of time there is no motion occurring. If
everything is motionless at every instant, and time is entirely
composed of instants, then motion is impossible.
In the Newtonian model of the universe, momentum/velocity is something
that objects have at every instant of time
I didn't know that. Perhaps it's why he could develop calculus reasoning with infinitesimals without addressing the philosophical problem and without formal notion of limits. His assumptions were not universally accepted at the time. The philosopher George Berkeley argued that
... forces and gravity, as defined by Newton, constituted "occult
qualities" that "expressed nothing distinctly". He held that those who
posited "something unknown in a body of which they have no idea and
which they call the principle of motion, are in fact simply stating
that the principle of motion is unknown."
( https://en.wikipedia.org/wiki/George_Berkeley#Philosophy_of_physics )
@leftaroundabout I agree that momentum is a better fundamental notion than velocity (certainly for quantum mechanics, possibly for Newtonian too). I don't think it's better to start calculus there, though.
@Hurkyl notes correctly that there are new mathematical structures - germs - that capture the idea of what happens near but not at a point. But I think the idea of the germ of a function is more technical and abstract than called for by the question.