I was reading this answer where I saw the following gif:
We can see that when a point becomes point of contact (i.e: touching the ground), the curve of it's motion has a cusp. To my knowledge, a cusp doesn't have a tangent line. So, how have physicists 'hacked' math to talk about velocity here?
As has been pointed out in comments, it may be possible to speak of a velocity at a point where the trajectory itself is not "smooth", if the velocity is zero at this point.
As a simpler example, consider the position of a particle given as a function of time by $$ \vec r = \alpha t^3 \hat x + \alpha |t|^3 \hat y.$$ The particle moves along the curve $y = |x|$, which has a "cusp" at $(0, 0)$. However, it can readily be shown that the position vector as a function of time is differentiable, i.e. the velocity is well defined everywhere. It is zero at the cusp (at $t = 0$).
As was pointed out in the comments, it is important to not confuse position versus time graphs and position versus position graphs. Velocity is the slope of the tangent line on a position versus time graph, not a position versus position graph.
If we graph $x$ versus time and $y$ versus time for a point on a wheel we see that both graphs have no cusps and in fact there are specific points in time where the tangent lines on both graphs has zero slope, corresponding to the instant where the point touches the ground.
The sinusoidal curve is the $y$ position of the point and the upward sloping curve is the $x$ position. The horizontal axis is time.
