Perhaps I need to explain it differently, since you don't want to accept that there are some matrices that completely lack an inverse.
Suppose you instead asked to compute the inverse of a number. So, for some X,find the number Xinv, such that X*Xinv==1.
Great, no problem, since we all know how to form the number Xinv=1/X. But then suppose I gave you X=0? That is, the one case where X lacks a multiplicative inverse? No matter what you do, there is no number Xinv, sich that 0*Xinv==1. Certainly you must accept that fact. (inf need not apply.)
The case of computing the inverse of a rank deficient matrix is the same. No inverse exists, when A is less than full rank. Wanting it to happen is not sufficient.
So then you ask, why is the pseudo-inverse not sufficient to form an inverse? But we just got done saying that if A is amatrix of less than full rank, then no matrix exists such that
A*Ainv == eye(size(A))
So even if we compute Ainv as the pseudo-inverse, it does not matter. We cannot get around the lack of a multiplicative inverse. The magic of an SVD is not sufficient, or even the fact it is called a pseudo-inverse. A name that sounds like it is an inverse is not sufficient to make it one.
What does the pseudo-inverse do then?
pinv(A), computed using the SVD, it is a computation that is nicely stable. When A has full rank, then pinv(A) should be the same as inv(A). And pinv(A) is a nice way to solve a linear system of equations, A*x=b, that is robust to singularity of the matrix A. A virtue of the pseudo-inverse built from an SVD is theresulting least squares solution is the one that has minimum norm, of all possible solutions that are equally as good in term of predictive value. I could probably list a few other properties, but you can read about them as easily in Wikipedia.
But it is not an inverse when A is singular.