This is a speculation and perhaps naive. The theorem of Siegel that

There exist only finitely many integral points on a curve of genus $\geq 1$ over a number ring $\mathcal O_{K, S}$ where $S$ is a finite set of places in a number field $K$

has proofs from arithmetic geometry, for instance by Parshin and Faltings, or by Kim. These proofs are interesting in that they use the modern developments in algebraic geometry and arithmetic to prove a theorem in number theory. But it is dissatisfying that as far as I am aware these proofs do not give an effective bound as opposed to the proofs using Baker's theorem. For instance, Silverman's book feels that the Parshin-Faltings proof is indirect.

But I do not know about the more recent developments and philosophy. I am wondering whether there is any hope that the more sophisticated proofs can be made effective, as it would give the best of both worlds. I hope the experts here can answer this.