A system utilizing uncountable ordinals better than OCFs?

Hi, first post. So this idea came to me as I was messing around with ordinal notation, being inspired by Stephen Brooks's transfinite number line, which was recently updated to include ordinals up to BHO. The concept is to ditch the multiple arguments of the extended Veblen function and instead use $\Omega$ to express the notion of transcending the said arguments. At first I thought it was also a valid OCF, but as I've now discovered, there are noticeable differences.

First of all, this is the main function, similar to the Psi function in OCFs, except made less powerful due to the absence of $\omega$ itself in the initial permitted notation: $\varphi(\alpha)=\omega^\alpha$.

This yields $\varphi (\Omega )=\varepsilon _{0}$ as the first ordinal that has been collapsed down from a higher uncountable one. Now though comes the difference from OCFs. If this were an OCF, the ordinals of the form $\varphi(\alpha)$would have been obtained through an inductive process and would cover the entire Wainer hierarchy after an infinite amount of steps. Similarly, since $\varepsilon _{0}\omega =\omega ^{\varepsilon _{0}+1}$ cannot be obtained by summing a finite amount of elements (such is true only for $\varepsilon _{0}n$; also note, addition is the only permitted binary operation here, as opposed to OCFs, which usually also include multiplication and exponentiation), this must therefore be the ordinal represented by $\varphi (\Omega +1)$.

However, this wasn't my initial idea, I instead thought of the phi notation as a substitute for the usual ordinal notation and arrived at the following equivalence instead: $\omega ^{\varepsilon _{0}+1}=\varphi (\varphi (\Omega )+1)$, which would be the usual way to apply the Veblen function actually. And, long story short, this leads all the way to $\Gamma _{0}=\varphi (\Omega ^{2})$ (and generally $\varphi (\Omega \alpha +\beta )=\varphi _{\alpha }(\beta )$ as far as the Veblen hierarchy is concerned), which can obviously be extended further. (In fact, it seems that $\Omega$ directly substitutes for new function arguments.)

So as you can see, there is potential in this kind of notation, being more economic with uncountable ordinals than OCFs usually are. However, I am unsure about the rigorousness of the definition of such a system; from what I've seen, OCFs are pretty solidly defined; this on the other hand is... different. Plus, I do not know if I am the first to come up with this. My guess is that probably not, considering the Wikipedia article about BHO already mentions something about using uncountable ordinals in the Veblen function (although without any examples). If such a system already exists, please share it, so that I, as they say, need not be reinventing the wheel over here.

Also, may someone please explain the meaning of $\Omega$ in Stephen Brooks's transfinite number line? One of the reasons I began exploring this concept is because he utilizes it for some ordinals after $\Gamma_{0}$, even though it doesn't look like an actual OCF (sometimes it for example directly substitutes for $\Gamma_{0}$). Again, if you have some sources for reading, do share them.