This question is a follow up to an earlier question of mine on enumerating elliptic curves of a given conductor.

I've heard people say that studying higher dimensional varieties via explicit defining equations often leads to hopelessly unmanageable complexity. As apparent evidence of this, on page 65 of Cassels' and Flynn's book titled Prolegomena to a Middlebrow Arithmetic of Curves of Genus $2$, the authors state that the defining equations that they find for Jacobian varieties of genus $2$ curves consist of $72$ quadratic equations in $\mathbb P^{15}$. People say that rather than studying higher dimensional algebraic varieties as solution sets to explicit equations, one typically studies such varieties in a more abstract and geometric way.

This makes sense. Yet I wonder how one can get look at concrete examples without defining equations. I know that there are some varieties such as moduli spaces which provide examples. But suppose, say, you want to prove that there are finitely abelian varieties of an arbitrary fixed dimension $d$ over a fixed arbitrary number field $K$ with a fixed conductor $N$ without looking at the automorphic side of things.

Is there a (conjectural) method of proving finiteness without writing down explicit defining equations?

[Edit: As Barinder Banwait points out, this follows immediately from the Shafarevich conjecture, which was proved by Faltings.]

Pushing the envelop further,

Is there a (conjectural) algorithm for enumerating these objects?

Pushing the envelop still further,

Suppose beyond enumerating such varieties, you want to determine, e.g., how many of them have surjective (mod $5$) Galois representation (say, attached to $H^1$) - is there a conjectural algorithm for doing so?