The answer is that you are dealing with approximations which are reasonable in some circumstances but not in others.
If you look at a derivation of the mirror formula $\frac 1 u + \frac 1 v = \frac 1 f = \frac 2 R$ you will always find an approximation made as shown in the diagram below.
However there is another flaw in the diagram in that the assumption which has been made is that all the rays parallel to the principal axis meet at the focal point $F$ where the focal length is half the radius of curvature of the mirror.
If you have ever looked at tea, coffee or water in a mug you will know that this meeting at a point for all parallel rays does not happen.
A caustic curve is formed.
This type of defect of a spherical mirror produced when the rays are not mear to the principal axis is called spherical aberration.
Correction is possible but not for all rays.
The most common type of correction is for use in reflecting telescopes where either a parabolic reflector is used or a aspherical reflector is used with a Schmidt corrector plate in front of it.
Even that does not completely eliminate spherical aberration as is shown below with parallel rays not parallel to the principal axis.
So the title to your question "Why do light rays intersect (or appear to intersect) at a specific point on reflection from spherical mirror?" is a good one in that it has in it the word "appear" which you can interpret as being related to the degree of accuracy of reproduction of the object as an image.
Have you ever looked at yourself in a magnifying shaving/make-up mirror and notices the distortion?