We present a universally-optimal distributed algorithm for the exact weighted min-cut. The algorithm is guaranteed to complete in $\widetilde{O}(D + \sqrt{n})$ rounds on every graph, recovering the recent result of Dory, Efron, Mukhopadhyay, and Nanongkai~[STOC'21], but runs much faster on structured graphs. Specifically, the algorithm completes in $\widetilde{O}(D)$ rounds on (weighted) planar graphs or, more generally, any (weighted) excluded-minor family. We obtain this result by designing an aggregation-based algorithm: each node receives only an aggregate of the messages sent to it. While somewhat restrictive, recent work shows any such black-box algorithm can be simulated on any minor of the communication network. Furthermore, we observe this also allows for the addition of (a small number of) arbitrarily-connected virtual nodes to the network. We leverage these capabilities to design a min-cut algorithm that is significantly simpler compared to prior distributed work. We hope this paper showcases how working within this paradigm yields simple-to-design and ultra-efficient distributed algorithms for global problems. Our main technical contribution is a distributed algorithm that, given any tree $T$, computes the minimum cut that $2$-respects $T$ (i.e., cuts at most $2$ edges of $T$) in universally near-optimal time. Moreover, our algorithm gives a \emph{deterministic} $\widetilde{O}(D)$-round 2-respecting cut solution for excluded-minor families and a \emph{deterministic} $\widetilde{O}(D + \sqrt{n})$-round solution for general graphs, the latter resolving a question of Dory, et al.~[STOC'21]