I read the weakly Mahlo ordinal is weakly inaccessible , hyper-weakly inaccessible, hyper-hyper-weakly inaccessible, (1@α)-weakly inaccessible, and so on as far as you diagonalize.
But is it the first cardinal with this property?
More concretely if you "define" $$a_0(x)=x+1$$ $$a_\alpha(x)=\text{the $x^{th}$ ordinal in } \{y\mid\gamma<y,\beta<\alpha,a_\beta(\gamma)<y,y\text{ is regular if $x$ is a succesor}\}$$ $$\text{if }\operatorname{cf}(\alpha)<N$$ $$a_\alpha(x)=a_{\alpha[x]}(x)\text{ if }\operatorname{cf}(\alpha)=N$$ $$N=\min\{y\mid \gamma<y,\operatorname{cf}(\beta)\le N,a_\beta(\gamma)<y\}$$
$N$ is the smallest ordinal unreachable using inaccessibility. But is this the Mahlo ordinal?
What if you add the restriction $N$ is regular?
If it is not the first Mahlo ordinal could you use it (with regularity) in a Mahlo OCF?
Analysis of $a$
$a_1(x)=\omega_x$
$a_2(x)=I_x$
$a_{2+\alpha} : \alpha\text{-inaccible}$
$a_N(2+x)=a_{2+x}(x)=\text{ the ${2+x}^{th}$ $x$-inaccessible}$
$a_{N+1} : (1,0)\text{-inaccible}$
$a_{N+N+1} : (2,0)\text{-inaccible}$
$a_{N\cdot\alpha+1} : (\alpha,0)\text{-inaccible}$
$a_{N^2+1} : (1,0,0)\text{-inaccible}$
$a_{N^2\cdot\alpha+1} : (\alpha,0,0)\text{-inaccible}$
$a_{N^3+1} : (1,0,0,0)\text{-inaccible}$
$a_{N^\alpha +1} : (1@\alpha)\text{-inaccible}$
It's clear $N$ is used as a diagonaliser, so $N$ is self must be larger than any diagonalisation of $I$, like $M$
I can't say I completely follow your definition, but you might want to look at the following paper:
Carmody, Erin, Killing them softly: degrees of inaccessible and Mahlo cardinals, Math. Log. Quart. 63: 256--264, Article ID https://doi.org/10.1002/malq.201500071, (2017).
She develops a useful notation system for the degrees of inaccessibility (like hyper-inaccessible etc.) and, among other things, investigates the relationship of these to Mahlo cardinals (see especially Section 3). In particular, she shows that it is consistent to have a cardinal which has all the degrees of inaccessibility (describable in her notation) but no Mahlo cardinals at all.