That's 👆 not a tiny angle.
Doing it with a tiny angle doesn't change anything.
Even with a tiny angle, you still have the tangent being a corner like that, with the arc cutting that corner to produce a shorter path, and the sine cutting straight across being the shortest path.
That means the arc length must always be between the sine and the tangent. There is no other option.
Literally all going to a smaller angle does is make the picture thinner or longer.
It does not change the underlying argument.
You can do the same thing by taking that picture and making it 1 million times longer, but the same width.
That will correspond to a much smaller angle for a much larger circle.
The tangent will still be a corner.
The arc will still cut that corner, making the path shorter than the tangent.
And the sine will still be a straight line, making it the shortest path.
If you want to refute this, you need to provide an actual argument, not just a baseless claim that the arc needs to be longer.
And no, saying there is extra length between the sine and tangent (i.e. that horizontal line in the diagram) does not mean the arc need to include that length.
Especially when we can already see it doesn't do that for larger angles.
e.g. for 90 degrees, with a radius of 1, the tangents forms a 90 degree corner, with a total length of both of 2.
The straight line from the sin is sqrt(2).
The distance between the straight line and the corner is sqrt(2)/2.
So the sum you claim the arc length needs to be would be sqrt(2)+sqrt(2)/2.
That is 3*sqrt(2)/2, which is roughly 2.12.
That is already longer than the corner.
That means you would already need pi to be over 4.
Your claim is pure garbage.
That width between the straight line and corner does NOT mean that the arc needs to be longer than the straight line by that much.