What HLD Cannot Do

Constant time LCA. HLD needs $O(\log n)$ to climb between chain heads, while findroot and the LCA are $O(1)$ in an LCT, so ancestor queries come free between updates.

The forest itself can change. HLD fixes the tree shape at build time, while link and cut are $O(\log n)$ amortized in an LCT, so edges come and go between queries.

Path aggregates at all. HLD lays its chains out in DFS order, so it answers only subtree queries, while path queries are $O(\log n)$ amortized in an LCT and come out of one splay tree.

A worst case bound. HLD costs $O(\log^2 n)$ amortized per path query, while link and cut are $O(\log n)$ worst case in an LCT, so no single operation can spike.