Here is why Universes is a "safe" assumption. Suppose it actually is consistent. Then it cannot possibly prove any arithmetical statement that contradicts an arithmetical theorem of PA or ZFC. This is because it proves that PA and ZFC have "internally standard" interpretations, and so everything that they prove about the naturals is true of the "standard" model (i.e. within the theory ZFC+Universes), and thus agrees with everything "true" (true according to ZFC+Universes).

If we further assume that ZFC+Universes has an actually standard model, then this means that all its arithmetical theorems are actually true. But this kind of begs the question, I think. We would like to know about some kind of coherency between various theories at a syntactical level, that they don't prove contradictory arithmetical statements.

The idea, I believe, is that if we can organize them linearly in a "standard interpretation hierarchy," then we are fine as long as we believe in the consistency of the strongest theory under consideration. To be more precise, we look at theories $T$ which have a "natural numbers object" $\mathbb N^T$, which should at least satisfy PA, and should probably be proven by $T$ to satisfy second-order PA. If $S$ is another such theory, and $T$ proves that $S$ has a model $\frak A$ such that $\mathbb N^{\frak A} = \mathbb N^T$, then $T$ and $S$ cannot disagree about what their respective natural numbers object satisfies.

This situation applies to large cardinal axioms more generally.

Let us abbreviate ZFCU = ZFC + "Every set is contained in a Grothendieck universe." Here is why ZFCU is a "safe" assumption. Suppose it actually is consistent. Then it cannot possibly prove any arithmetical statement that contradicts an arithmetical theorem of, for example, $n^{th}$-order PA for any $n$. This is because it proves that $n^{th}$-order PA has an "internally standard" interpretation, and so everything that it proves about the naturals is true of the "standard" model (i.e. within the theory ZFCU), and thus agrees with everything "true" (true according to ZFCU).

If we further assume that ZFCU has an actually standard model, then this means that all its arithmetical theorems are actually true. But this kind of begs the question, I think. We would like to know about some kind of coherency between various theories at a syntactical level, that they don't prove contradictory arithmetical statements (or other statements about "standard objects").

The idea, I believe, is that if we can organize them linearly in a "standard interpretation hierarchy," then we are fine as long as we believe in the consistency of the strongest theory under consideration. To be more precise, we look at theories $T$ which have a "natural numbers object" $\mathbb N^T$, which should at least satisfy PA. If $S$ is another such theory, and $T$ proves that $S$ has a model $\frak A$ such that $\mathbb N^{\frak A} = \mathbb N^T$, then $T$ and $S$ cannot disagree about what their respective natural numbers object satisfies.

I don't at present have an all-encompassing definition of "standard model," but here are some examples to illustrate the idea.

Example: ZFC proves that for all $n<\omega$, $\mathbb N$ satisfies $n^{th}$-order PA.

Example: ZFCU proves that for each $n<\omega$, there is a standard model of $I_n = $ ZFC + "There are exactly $n$ inaccessible cardinals." This should be the least rank $V_\alpha$ satisfying this theory, so that it is definable. Although the $I_n$ are mutually inconsistent and inconsistent with ZFCU, whatever $I_n$ proves, ZFCU proves it holds in $I_n$'s standard model. $I_n$ has a standard model of $I_m$ for $m<n$. Iterating the operation of taking standard models "commutes" in some sense.

Non-example: ZFCU proves the consistency of $I_1$ + $\neg Con(I_2)$. However, it proves that this does not hold in the standard model.