How much of gravity is caused by time dilation?

The video is just nonsense. Early on, he starts talking about "curvature in time" and saying "time can curve." He just has no idea what he's talking about. Curvature is curvature of spacetime, not time or space separately.

Gravitational time dilation isn't generically even something that can be defined in general relativity. It can only be defined for certain kinds of spacetime that are called static. What is true is that in such a spacetime, the time dilation is closely related to the gravitational potential. And even in this special case, it's not a question of "causing" gravity.

Do yourself a favor and stop trying to learn about science from videos. Read a book.

We perceive gravity as a fictitious force as we are pushed off our geodesic by the surface of the Earth, according to the geodesic equation (the version that uses time):

$$ \ddot x^{\mu} = -\Gamma^{\mu}_{\alpha\beta} \dot x^{\alpha}\dot x^{\beta} + -\Gamma^0_{\alpha\beta} \dot x^{\alpha}\dot x^{\beta}\dot x^{\mu}$$

The point of this equation, here, is to show that it's not just time curvature that matters. Alpha and beta run over all the indices.

However, when you are on the Earth's surface sitting at rest:

$$ \dot x^{\mu} \approx (c, 0, 0, 0) $$

and $\alpha$ (or $\beta$) $= 1,2,3$ just don't get an opportunity to enter the calculation.

Also, when you plug in Christoffel symbols for the Schwarzschild metric at $r\gg r_s$, you get the nice result that:

$$ \ddot x^r = -\frac{GM}{(x^r)^2} $$

which is Newton's law of gravitation.

If you get closer to the Schwarzchild radius, move at relativistic velocities, or give the gravitating body lots of angular momentum, then you need more terms than $\Gamma^r_{tt}$ to quantify gravitation.

Back in the weak field of Earth, you can imagine time curvature by considering that path dependence of time as follows:

Two researchers (A & B) in the basement at NIST have atomic clocks, and agree to meet in the top floor conference room in 1 year. A takes the elevator up and waits in the conference room, B sits in the basement for a year, and only then takes the elevator up top. He sees A who then says, "You're late"....their two right angle paths do not meet at the same point in spacetime.

How much of the attractive "force" (well, it is not actually a force) felt by an object in the gravitational field of another object like a planet or star is caused by clocks closer to the object runnning slower (compared to clocks further away) compared to the object moving toward the heavier object due to curved geodesics?

The geodesics are paths in the 4 dimensional spacetime, not in 3-D space only. They include time effects.

The video tries to explain the accelerated movement associated to gravity to differences in the clock rate at different point of an object. It is not the case.

Suppose an object thrown upward. Its velocity will decrease gradually until stops, and then changes direction, and gradually increases until it has the same initial speed, (but opposite direction).

Compare that movement with a plane flying from Paris to Seattle (both cities with similar latitudes about $48^\circ$. Looking at a globe, it is clear that the shortest path (that is part of a great circle, called geodesic) requires keeping a north component of the velocity, that gradually becomes smaller until reach a maximum latitude (about $64^\circ$). Then a small south component appears, that are getting greater until reach Seattle.

If we compare with the previous example, taking latitude as height and longitude as time, the situation is similar to the object thrown upward.

For the plane, the direction of the velocity changes during all the trip due to the curvilinear coordinates. So the movement could be called accelerated. But we know that it is as straight as possible.

For the object, the path is also as straight as possible (a geodesic), but it is shown to us as accelerated, due to the curvilinear coordinates.