I totally agree. This is why I wrote "sketch of a proof" instead of "proof".
Besides, I've left a case when there is one large circle (L) and a number of small circles,
small circles are pair-wise externally tangent to each other and each small circle is internally tangent to the big circle.
I just said, that "it is easy to see" that at most 3 small circles are possible, so I just want to complete it.
Suppose we have 4 or more inner circles. Let's number them 1, 2, 3, 4... in the following way:
number 1 is assigned freely, and subsequent numbers are assigned subsequently clockwise according to tangent points to the circle L.
Since circle 2 is tangent to circle 4 and both are tangent to the circle L, they divide interior of circle L in two parts.
But circles 1 and 3 are in different parts, so these circles cannot be tangent.
This contradiction shows that 4 or more small circles are not possible.