The small magnitude of velocity at the root does not dictate whether the section there is stalled or not.
The local angle of attack determines whether the flow is stalled. The magnitude of velocity at the root helps determine the local angle of attack.
The FAA document is trying to represent a standard diagram from the theory of the blade element method -- but they've added so much color fade and un-related stuff that it actually makes it more difficult to understand in my opinion.
Here is another example of the same diagram -- search for 'blade element theory' to find more.
Unfortunately, this image is rotated and flipped relative to the FAA one, so I've manipulated it to be in the same orientation here.... This makes the labels read like a mirror, but hopefully you can use this to convince yourself that it is depicting the same thing as the FAA version, then we can work with the one where the labels are readable.
It is helpful to talk about velocity around a propeller in three components. Instead of x,y,z components, we will use cylindrical components. Think of a cylinder whose axis is aligned with the propeller axis and the direction of flight. The circular outside of the cylinder corresponds with the tips of the spinning propellers. We will call the three components of velocity:
Axial -- along the cylinder axis, parallel to the axis of rotation of the prop, aligned with the direction of flight.
Radial -- along the cylinder radius, from the root of the blade towards the tip of the blade.
Tangential -- tangent to the motion of the blade, this direction follows a curved path, like the grooves on a record.
This diagram is drawn in the Axial-Tangential plane. The radial direction would be coming out of the paper and is not depicted.
The FAA diagram also ignores the $w_a$ and $w_t$ contributions (in cyan). We can ignore them for now too.
The forward velocity of the aircraft (airspeed) is $V_\infty$. This is the primary axial component of the velocity.
The rotational speed of the propeller is $\omega$. It gets multiplied by the radius $r$ to give the tangential component of velocity $u=\omega r$.
Near the center ($r$ is small), the tangential component is small. Near the tip, ($r$ is large), the tangential component is large.
Together, $V_\infty$ and $u$ add together through vector addition to form $w$, the velocity each blade section 'sees'.
Although every blade section sees the same $V_\infty$, the variation of $r$ from root to tip means that the magnitude and angle that $w$ makes with the plane of rotation varies from very steep at the root (where $r$ is small) to very shallow at the tip (where $r$ is large). This angle is labeled $\phi$ (it is not given a label in the FAA diagram).
The geometric blade pitch (measured relative to the plane of rotation) is $\beta$.
If we subtract the local wind angle $\phi$ from the blade pitch $\beta$, we get the local angle of attack $\alpha=\beta-\phi$.
So now we can tell that the local angle of attack of a blade section depends on the flight condition ($V_\infty$ and $\omega$) and also on the blade twist $\beta$.
We know that $\phi$ will be large at the root and small at the tip.
If $\alpha$ gets too big, the blade will stall.
The FAA mentioned an un-twisted blade (like a ceiling fan blade). In such a blade, the twist will be constant $\beta=Constant$. Lets assume $10^\circ$. In that case, considering the variation of $\phi$, we are forced to have $\alpha=10-\phi$. At the root, $\phi$ is large, so $\alpha$ might go negative. At the tip, $phi$ is small, so $\alpha$ might be large enough to stall the airfoil.
One idea for designing a propeller might be to seek to achieve constant $\alpha$ (real props are more sophisticated than this, but this is good to the first order). So if you want to achieve constant $\alpha$ (say $3^\circ$), and we know $\alpha=\beta-\phi$, then we will need $\beta$ to follow $\phi$ in behavior -- large at the root, small at the tip.
