It's not as easy to visualize the tangent magnitude as you are suggesting, because the time parametrization makes the calculation fail - the curve doesn't travel forward at a constant rate. This is best seen with the second example: If the left handle is fully extended and the right handle is 0, the Bézier curve looks almost exactly the same as when the handles are reversed. Here's a picture: https://i.imgur.com/WkanN1G.png
The handle varies from 0 to full-strength, but the magnitude of the tangent vector stays constant. This means the handle doesn't decide the magnitude. Tracing the path in your head to visualize the changing tangent vectors would mean visualizing a competition between t^2, (1-t)^2t, and (1-t)^3, which I find difficult, even with some calculus knowledge.
The handle varies from 0 to full-strength, but the magnitude of the tangent vector stays constant. This means the handle doesn't decide the magnitude. Tracing the path in your head to visualize the changing tangent vectors would mean visualizing a competition between t^2, (1-t)^2t, and (1-t)^3, which I find difficult, even with some calculus knowledge.