I don't have anything to contribute on the philosophical questions here; instead, I'll look at this:

It also seems like this model is "maximal" in the sense that if the existence of a set $A$ doesn't violate any of the axioms of ZFC, then $A$ exists in the universe of sets.

Unfortunately, this idea doesn't work; we can see why with the Continuum Hypothesis.

In a model where CH is true, then by definition we have a bijection $\phi_1 : 2^{\aleph_0} \leftrightarrow \aleph_1$. That's what it means for the two cardinals to be equinumerous. Also, remember that everything—including functions—is a set. In particular, $\phi_1$ is just a set of ordered pairs: it's a subset of $2^{\aleph_0}\times\aleph_1$ demonstrating a 1-to-1 correspondence between the two. In other models, ones where CH doesn't hold, $2^{\aleph_0}$ will be equal to some other cardinal number.* E.g., there's a model where we have a bijection $\phi_2 : 2^{\aleph_0} \leftrightarrow \aleph_2$.

See the problem? $\phi_1$ is a set that exists in some models, and $\phi_2$ is a set that exists in others, but they can't both exist in the same model. Otherwise, we'd just compose the two to get a bijection proving $\aleph_1 = \aleph_2$, which is provably false (and therefore false in every model). So there can't be a "maximal model", because despite each set being valid in some model (and so both on their own are consistent with ZFC), no model can contain both.

This is a case of a more general phenomenon: models can really screw with your intuition of what it means for a statement to be "true". My favorite example is the fact that the set of von-Neumann naturals $\omega = \{0,1,2,\ldots\}$ (where each ordinal is the set of those less than it) doesn't model ZFC (e.g., it doesn't satisfy the axiom of pairing), but it DOES satisfy the powerset axiom! This is despite the fact that the "real" powerset of $2 \in \omega$ is $\mathscr{P}(2) = \mathscr{P}(\{0,1\}) = \{\{\},\{0\},\{1\},\{0,1\}\} \notin \omega$. See the linked article for an explanation of how this possibly makes sense. Basically, inside the model, $\mathscr{P}(2) = \{\{\},\{0\},\{0,1\}\} = 3$. While in "reality", $\{1\} \subset \{0,1\}$, inside the model the set $\{1\}$ doesn't even exist (and so in particular asking whether it's a subset of anything isn't even a well-formed question).

The bottom line is that the perspective inside a model doesn't necessarily match the perspective inside another model or from the "outside". This is because statements like "$X$ and $Y$ are equinumerous" or "$X$ is the powerset of $Y$" are actually statements not just about $X$ and $Y$, but about other sets in the universe (namely, a bijection $\phi$ between $X$ and $Y$, and subsets $S$ of $X$), and those other sets may or may not exist in the model alongside $X$ and $Y$.

* Actually, showing that every cardinal number is an aleph number requires the axiom of choice. But that doesn't affect the argument, because every model of ZFC is also a model of ZF; as such, the same two contradictory models still exist without Choice.

I don't have anything to contribute on the philosophical questions here; instead, I'll look at this:

It also seems like this model is "maximal" in the sense that if the existence of a set $A$ doesn't violate any of the axioms of ZFC, then $A$ exists in the universe of sets.

Unfortunately, this idea doesn't work; we can see why with the Continuum Hypothesis.

In a model where CH is true, then by definition we have a bijection $\phi_1 : 2^{\aleph_0} \leftrightarrow \aleph_1$. That's what it means for the two cardinals to be equinumerous. Also, remember that everything—including functions—is a set. In particular, $\phi_1$ is just a set of ordered pairs: it's a subset of $2^{\aleph_0}\times\aleph_1$ demonstrating a 1-to-1 correspondence between the two. In other models, ones where CH doesn't hold, $2^{\aleph_0}$ will be equal to some other aleph number.* E.g., there's a model where we have a bijection $\phi_2 : 2^{\aleph_0} \leftrightarrow \aleph_2$.

See the problem? $\phi_1$ is a set that exists in some models, and $\phi_2$ is a set that exists in others, but they can't both exist in the same model. Otherwise, we'd just compose the two to get a bijection proving $\aleph_1 = \aleph_2$, which is provably false (and therefore false in every model). So there can't be a "maximal model", because despite each set being valid in some model (and so each on their own is consistent with ZFC), no model can contain both.

This is a case of a more general phenomenon: models can really screw with your intuition of what it means for a statement to be "true". My favorite example is the fact that the set of von-Neumann naturals $\omega = \{0,1,2,\ldots\}$ (where each ordinal is the set of those less than it) doesn't model ZFC (e.g., it doesn't satisfy the axiom of pairing), but it DOES satisfy the powerset axiom! This is despite the fact that the "real" powerset of $2 \in \omega$ is $\mathscr{P}(2) = \mathscr{P}(\{0,1\}) = \{\{\},\{0\},\{1\},\{0,1\}\} \notin \omega$. See the linked article for an explanation of how this possibly makes sense. Basically, inside the model, $\mathscr{P}(2) = \{\{\},\{0\},\{0,1\}\} = 3$. While in "reality", $\{1\} \subset \{0,1\}$, inside the model the set $\{1\}$ doesn't even exist (and so in particular asking whether it's a subset of anything isn't even a well-formed question).

The bottom line is that the perspective inside a model doesn't necessarily match the perspective inside another model or from the "outside". This is because statements like "$X$ and $Y$ are equinumerous" or "$X$ is the powerset of $Y$" are actually statements not just about $X$ and $Y$, but about other sets in the universe (namely, a bijection $\phi$ between $X$ and $Y$, and subsets $S$ of $X$), and those other sets may or may not exist in the model alongside $X$ and $Y$.

* Actually, showing that every cardinal number is an aleph number requires the axiom of choice. But that doesn't affect the argument, because every model of ZFC is also a model of ZF; as such, the same two contradictory models still exist without Choice.