I think the answer is "probably not." The reason is that projective space has *two* universal properties which are used to prove different kinds of things about it. One of these is the slick universal property you like, and the other is the clunky one which results in unpleasantries.

Though each universal property implies the other (since it uniquely identifies projective space), it seems unlikely to me that you can effectively do anything if you try to avoid one of them altogether.

One universal property makes it easy to understand maps *to* projective space:

$$
Hom(T,\mathbb P^n) = \{\mathcal O_T^{n+1}\twoheadrightarrow \mathcal L| \mathcal L\text{ a line bundle}\}
$$

Without bending over backwards (i.e.~reproducing the usual theory), I'd be surprised if you could use this universal property to even prove that there are no non-constant regular functions on $\mathbb P^n$.

I expect constructions that naturally pull back along morphisms (e.g. line bundles, regular functions) to behave like morphisms *from* projective space, so it would be strange if you could attack such constructions with this universal property.

Another universal property makes it easy to understand maps *from* projective space: $Hom(\mathbb P^n,T)$ is the equalizer of the two restriction maps $Hom(\coprod_{i=0}^n \mathbb A^n,T)\rightrightarrows Hom(\coprod_{i,j}\mathbb A^{n-1}\times (\mathbb A-0),T)$.

I guess this is the one that you don't like, but we're lucky to have it since it actually makes it possible to make sense of projective space having Zariski local properties (e.g. being smooth, $n$-dimensional, etc.), and thereby makes it possible to do geometry on it.