Suppose $M$ is a regular compact submanifold with boundary of dimension $n$ in $\mathbb R^{n+1}\smallsetminus 0$, and assume that for each $v\in S^n$ the ray emanating from $v$ intersects $M$ in at most one point. Denote by $S$ the subset of $S^{n}$ where this happens. The surjective submersion $r: \mathbb R^{n+1}\smallsetminus 0\longrightarrow S^n$ restricts to a bijection $r : M\longrightarrow S$.
Note that $\ker r_{p,\ast}=\langle p\rangle$ intersects $M$ at one point, but it can happen that $\langle p\rangle \subseteq T_pM$ (for example, draw a projected $x\mapsto x^3$ around $S^1$). However, in the one dimensional case, there are only finitely many such points where this fails, and one can perturb $M$ by a diffeomorphism in a neighbourhood of such points to preserve all hypothesis and guarantee that $r$ restricts to a submersion $r: M\longrightarrow S$ which must then be a diffeomorphism.
Suppose now that $n>1$. One can mimic the above to obtain a submanifold $M$ where the points where $\langle p\rangle \subseteq T_pM$ form a $n-1$ dimensional submanifold in $S^n$. For example, by extending the previous example in higher dimensions, one can produce a circle inside $S^2$, etc.
Can one in general modify $M$ slightly by a diffeomorphism so that $\langle p\rangle + T_p M =T_p\mathbb R^{n+1}$?
This is not an answer but an extended comment. Start with the case $n=1$ when $M$ is diffeomorphic to $[0,1]$. Via polar coordinates, the problem becomes the one about smooth curves in $R^2$ which are graphs of continuous monotonic functions $f: [a,b]\to R$: The projection $r$ corresponds to the projection to the $y$-axis. Note that $f$ itself need not be smooth as the tangent to the graph could be vertical. In any case, a continuous monotonic function can indeed be approximated by smooth monotonic functions with nonvanishing derivatives and hence, the projections from their graphs to the $y$-axis become submersions. However, the original function $f$ can have an arbitrary measure zero Cantor set $C$ as the set of its critical points, i.e. points where $f'(x)=0$. Thus, the claim you make about finiteness of the set of points in $M$ where $dr=0$ is false.
The situation becomes much more complex in higher dimensions. I will think in terms of perturbing an injective smooth map $r: M^n\to N^n$ (where in your case $N^n=S^n$). I see no way to find a perturbation such that the critical set of the perturbed map is a submanifold and the new map is still 1-1. Once you loose injectivity, you easily get a "fold singularity" which, in turn, cannot be perturbed away. A cautionary example is the one of Milnor's exotic spheres $\Sigma^n$ which admit smooth injective maps $\Sigma^n\to S^n$. However, unlike in your case, the derivative vanishes identically at a point in these examples.
On the other hand, once you have a smooth injective map $f: M^n\to N^n$ whose derivative has rank $\ge n-1$ evereywhere and the singular set is a codimension 1 submanifold, then $f$ can be approximated by immersions.