In the Grothendieck universe approach to category theory, as you say, we replace all small sets with $\mathcal{U}$-small sets. Let's look at the definition of a locally presentable $\mathcal{U}$-category, for example, since that's in the conclusion of the theorem you mentioned: a locally presentable $\mathcal{U}$-category is a category with $\mathcal{U}$-small homsets which has all $\mathcal{U}$-small colimits and for which there exists a $\mathcal{U}$-small regular cardinal $\lambda$ and a $\mathcal{U}$-small set of $\lambda$-compact objects which generate the category under $\lambda$-filtered $\mathcal{U}$-small colimits. The size relationships here are key. For example, the category of *all* sets is not a locally presentable $\mathcal{U}$-category because its hom-sets are not $\mathcal{U}$-small, while the category of sets of cardinality less than some $0 < \kappa < \mathcal{U}$ is not a locally presentable $\mathcal{U}$-category because it lacks $\mathcal{U}$-small colimits.

Now one formulation of Vopěnka's principle is

There is no proper class-sized discrete full subcategory of $\mathrm{Gra}$ (some category of graphs)

or in a more positive form

Any discrete full subcategory of $\mathrm{Gra}$ has only a set of objects

The proof of a theorem like the one you mentioned (in the non-universe setting) is going to go something like this. We use this axiom to conclude that some particular transfinite construction of new objects in a category cannot go on "forever". It has to stop at stage $\gamma$ for some ordinal $\gamma$, and from $\gamma$ we'll somehow produce the cardinal $\lambda$ in the definition of a locally presentable category and the set of generators.

If we now adopt universes, you may be thinking that

Any discrete full subcategory of $\mathrm{Gra}_{\mathcal{U}}$ has only a set of objects

is just (trivially) *true*, even without Vopěnka's principle. But if we tried to repeat the above argument using $\mathcal{U}$-categories, we'd just learn that there is some $\gamma$ and $\lambda$ as before, but we have no control over their sizes. For example, maybe the $\lambda$ we end up with would be larger than the cardinality of our category and the set of generators just all the objects of our category. In order to show that $\lambda \in \mathcal{U}$ we would need

Any discrete full subcategory of $\mathrm{Gra}_{\mathcal{U}}$ has only a $\mathcal{U}$-small set of objects

and I guess this is what it means for $\mathcal{U}$ to be a Vopěnka cardinal, as Mike mentioned in a comment. If you believe in such cardinals then you believe in Con(ZFC + Vopěnka’s principle), so you are still relying on set-theoretic assumptions (and much more so than simply believing in universes in the first place).