This is only a partial answer that I cannot see how to complete easily, but may help others finding one. Applying a stereographic projection with center the north pole $P$ onto a plane parallel to the tangent plane at $P$ (for instance the plane of the equator), the circles that pass throught $P$ become precisely the lines of the plane. Therefore the question gives rise to the following one:
Suppose a finite set $S$ of lines the plane has the property "whenever two lines of $S$ intersect, there is at least a third line of $S$ that passes through the point of intersection". Must it then be the case that the lines of $S$ either are all be parallel or all pass through one fixed point of the plane?
The condition added to the original question (non-tangency of circles at $P$) amounts to excluding parallel lines in the set (whose circles would be tangent at $P$); however I believe the even the question above to have an affirmative answer, which is a stronger statement (one can allow parallelism, provided one just claims that there cannot be more than one point of intersection between some lines of $S$). I think this should even be a fairly classical result, but I could not see any simple argument (based on counting lines and interections, or on considering the convex hull of all intersection points, and such) that proves it. Probably I'm overlooking something simple.
One can however get fairly close to a configuration that would contradict the property, for instance the sides and diagonals of a paralellogram, which give just one forbidden intersection of only two lines (at the center). Or a triangle with its medians and the lines joining the midpoints of the sides (there are three spurious intersections). Of course there are infinite collections of lines that disprove the statement without "finite", for instance the lines in a three directions on a triangular grid or even simpler the set of all lines of the plane.
Since almost-counterexamples often involve parallel lines, it might be easier with the extra assumption of no parallel lines in $S$, but I still can't see an easy proof.