I think you're misunderstanding the theory Boolos describes (and so I disagree with your claim "I think this clearly captures the ranking function and Stage theory Boolos is speaking about"). When he writes

what he means is (I think) "if $\varphi$ is a formula associating each set to at least one stage, ..." - this is why the resulting principle is really a scheme ("We could have taken as axioms all instances of a principle which may be put ..."). That is, the "no matter how" means that the principle ranges over (definable) means of correlation.

This is fundamentally different from simply adjoining a ranking function to the universe. For example - and supposing we've already shown that "everything up to the least $\omega_-$-fixed point exists" - consider the formula $\varphi$ sending a set $x$ to $\omega$ if $x$ is infinite or empty, and to $\omega_{\omega_{\omega_{...}}}$ ($\vert x\vert$ times) if $x$ is finite and has at least one element. Boolos' principle applied to this formula then says that there is a stage by which each $\omega_{\omega_{...}}$ (finitely many times) has entered, which implies that the least fixed point of the $\alpha\mapsto\omega_\alpha$-function exists.

This theory does indeed imply full replacement, so long as the class of all sets which appear by stage $s$ is a set for every stage $s$ (note that this principle is not actually a consequence of ZC alone, so some replacement is already snuck in): given an instance of replacement, we can simply find a large enough stage such that witnesses to every "input" have already appeared, and then apply separation to the corresponding level of the cumulative hierarchy.

EDIT: I've rewritten for clarity.

First, re: your claim "It appears that what Boolos is saying is that: when we extend the rough iterative conception of set with a ranking function, then we get Replacement," this is incorrect, or at least incomplete. Boolos' principle is basically just saying "$Ord$ is regular," which when combined with just a small bit of replacement (which Boolos has already baked in) gives full replacement.

The key point is phrasing the hypothesis correctly. When Boolos writes

this is just saying "For every class relation $R\subseteq Sets\times Stages$ such that $dom(R)$ is all of $Sets$, ..."

This makes his whole proposed principle simply:

Suppose $R\subseteq Sets\times Stages$ is a class relation such that $dom(R)$ is all of sets. Then for every set $z$, there is some stage $s$ such that for each $w\in z$ there is a stage $t$ with $(i)$ $t$ earlier than $s$ and $(ii)$ $wRt$.

ZFC satisfies this principle by taking $$s=\sup\{\min(R^{-1}(w)): w\in z\}+1,$$ which exists by Replacement. Conversely, we can use Boolos' principle to prove Replacement by using it to find a sufficiently large stage that all witnesses have appeared with respect to the usual rank notion, and then applying separation. In gory detail:

• Suppose we have an instance of replacement: that is, a set $z$ and a formula $\varphi$ (with parameters) such that for each $w\in z$ there is exactly one $y$ with $\varphi(w,y)$.

• Let $rank$ be the usual ranking function on the universe of sets. Consider now the following relation $R$:

  • If $w\not\in z$, we set $wR\alpha$ for every ordinal $\alpha$.

  • If $w\in z$, we set $wR\alpha$ if the unique $y$ satisfying $\varphi(w,y)$ has $rank(y)=\alpha$.

• The relation $R$ satisfies the hypotheses of Boolos' principle (conflating ordinals and stages), and so that principle gives us some ordinal $\theta$ such that for each $w\in z$, the unique $y$ satisfying $\varphi(w,y)$ has $rank(y)=\theta$.

• Now consider $V_{\theta+1} = $ the sets of rank $\le\theta$ in the usual sense (the set-hood of $V_\theta$ needs to be justified, and as I commented below ZC alone isn't up to the job, but if I recall correctly the hypothesis that each "stage class" is a set is indeed built in). Because $\theta$ is large enough to see all the sets we care about appear, applying separation to $V_{\theta+1}$ we get that the class $\{y: \exists w\in z(\varphi(w,y))\}$ is a set. So we're done.

Actually, it's arguable that Boolos isn't identifying stages with ordinals at this, well, stage. However, that makes no difference: full replacement lets us conflate arbitrary well-orderings and ordinals, and Boolos' principle restricted to ordinals-as-stages gives full replacement as per the above.

Now, the theory you describe is quite different, and your comment on its limitations is correct: it's much weaker than ZFC. In particular, it holds in the structure $M=(L_{\theta+1},\in)$, where $\theta$ is the least fixed point of the map $\alpha\mapsto\omega_\alpha$. (I use $L$ instead of $V$ to control the length of well-orderings that show up; whether $(V_{\theta+1},\in)$ satisfies your theory is independent of ZFC, since we could have the continuum much larger than $\theta$.)

As to your philosophical critique of replacement, this is of course a somewhat subjective issue. I waffle on whether it's built into the cumulative hierarchy idea already; I tend to fall on the side of "yes," but that's not universal, and it seems Boolos takes the opposing position ("it does seem to us to be a further thought, and not one that can be said to have been meant in the rough description of the iterative conception"). I do certainly think that Replacement, and before that Infinity, do indeed constitute "ur-large-cardinal" principles. I believe Kanamori's article In praise of replacement backs majority-me against Boolos, but I haven't read it in a while so I can't promise it's fully on-topic (I do remember it being quite good though).