Some questions about the Hyperuniverse Program

Question

The Hyperuniverse Program, founded by Sy D. Friedman, intends to produce new second-order axioms of set theory which appropriately formalize "the universe is maximal" in one of a few ways. A height-maximality principle is one intended to formalize "the class of ordinals is as long as possible" (1, p.9). First-order axioms are not considered, because by the reflection principle, they do not effectively enforce this (1, p.14). As a first attempt for a height-maximality principle, the extended reflection axiom $\textrm{ERA}$ is introduced, then strengthened to obtain the principle of $\sharp$-generation - the latter "stands out as the correct formalization of the principle of height maximality" (1, p.17).

I have questions about the undertaking of the program:

1. Assume second-order reflection holds in $V$, which follows from $\textrm{ERA}$ and therefore from $\sharp$-generation. If $\phi$ is a second-order formula claimed to be an optimal height-maximality principle, if we assume $\phi$ then there exists $\alpha$ where $(V_\alpha,\in,V_{\alpha+1})\vDash\phi$. However, existence of a set-sized model satisfying $\phi$ is enough to rule $\phi$ out from being the optimal maximality principle - all first-order $\phi$ are ruled out because they have set models, leading to the immediately stronger height-maximality principle "$\phi$ and there is a set model of $\phi$". (I would argue the existence of a set-sized model of $\phi$ alone is enough, it shows $\phi$ fails to formalize "$\textrm{Ord}$ is as tall as possible".)
2. Notions from some areas of set theory, such as elementary embeddings, are forbidden for use in formulating maximality principles, as they "construe maximality with reference to, and in light of, specific theories". Principles only referring to "[focus] on the very properties of the cumulative hierarchy (of $V$) ...height and width" are preferred (2, p.6). However, as many of the strongest large cardinal axioms to date rely on elementary embeddings between classes, it seems like arbitrary elementary embeddings are accepted as natural statements inherently entwined with the statement "there are many ordinals". (Joel David Hamkins has called large cardinal axioms "genuine, robust mathematical concepts of infinity", unless large cardinals past measurable are excepted from this quote.)
3. About the preceding "general properties of $V$" requirement, height-maximality principles stronger than $\sharp$-generation are introduced, but they are ruled out by it: (2, p.12)

One can show that $\sharp$-generation implies all forms of reflection which are compatible with $V=L$ and, as a consequence of this ... it can legitimately claim to be the optimal principle expressing the height maximality of V. But why does it have to be so? Couldn't there be other (stronger) forms of reflection which imply $\lnot(V=L)$, and which also account for the existence of much stronger large cardinal hypotheses?

Mathematically, research in this direction has been conducted, and has, in particular, produced what we will call here embedding reflection. ... However, the use of and reference to arbitrary embeddings makes this principle incompatible with our maximality protocol, which, as we have seen in Sect. 2, just addresses 'general' features of V, and not specific theories or model-theoretic techniques.

These principles are strong enough to prove existence of supercompact cardinals, so they seem quite effective at enforcing "$\textrm{Ord}$ is as tall as possible". In which case, why restrict principles to reference general properties of $V$, if dropping the requirement seems to open the door for these very strong formulations?

4. Assume that some $\phi$ is an accepted formalization of height-maximality, e.g. the axiom of $\sharp$-generation. Let $\phi'$ be the axiom $\forall\alpha\exists(\beta>\alpha)((V_\beta,\in,V_{\beta+1})\vDash\phi)$. Is $\phi'$ ruled out on the grounds of "general properties of $V$"? If not, is $\phi'$ necessarily inconsistent with ZFC? Otherwise, would $\phi'$ be a strictly stronger height-maximality principle, as it enforces existence of many models where, when the universe is chopped at that model, the claimed height-maximality principle $\phi$ holds?

Edit: A shorter version of this question is posted on MathOverflow.

Sources:

1. Friedman, "Evidence for Set-Theoretic Truth and the Hyperuniverse Programme"
2. Friedman, Ternullo, "Maximality Principles in the Hyperuniverse Programme"