It's a bit long for a comment, but I'll make several points.
These are not uncommon ordinals.
I've seen them used in Rathjen's ordinal collapsing function involving Mahlo cardinals, which he denotes $\Phi$. As the comments point out, they appear in various places.
This is not at all how the multivariable Veblen function behaves (before the edit).
Your $\phi_{1,0}'(0)$ is simply $\phi_{\phi_1'(0)}'(0)$. It would be akin to saying that $\Gamma_0=\phi(\phi(1,0),0)$, which is not at all true.
To explain how the multivariable Veblen function works, I recommend seeing it as recursively closing over itself on lexicographically smaller arguments. In short, left-most arguments are more significant than right-most arguments. That is, we have things like $(1,0,0)>_L(\omega,0)>_L(3,0)>_L(2,\omega)>_L(1,0)$. From this, one can see that $\Gamma_0=\phi(1,0,0)$ is greater than $\phi(\alpha,\beta)$ for any $\alpha,\beta<\Gamma_0$. This can be shown to be equivalent to
$$\phi(1,0,0)=\sup\{\phi(1,0),\phi(\phi(1,0),0),\phi(\phi(\phi(1,0),0),0),\dots\}$$
but makes more sense when considering transfinitely many arguments.
As far as I can tell, it's significantly smaller than the usual Veblen function modified with $\phi(\alpha)=\omega_\alpha$.
The Veblen function is already optimal, as far as this kind of recursion goes. Thus, the fact that your functions have significantly less arguments than the general Veblen function will make it much smaller. A quick look and I'd say only 5 or 6 arguments of the Veblen function would be needed to outperform your functions.