Simple proof that splay trees have the dynamic finger property?
There has been a lot of further work more broadly on Splay trees and related data structures after the two papers that you mention, for example the 2019 paper of Levy and Tarjan [1].

However, there have been no alternative proofs or simplifications of the dynamic finger result of Cole et al. that you mention. (Any such attempt would be quite interesting to those working on these problems.)

There is, however, a 2016 paper by Iacono and Langerman [2] that proves a generalization of the dynamic finger property for a different data structure called Greedy BST, that matches many (all) known properties of Splay. The proof techniques here are easier and quite different from those in the Splay dynamic finger paper.

[1] https://doi.org/10.1137/1.9781611975482.80

[2] https://arxiv.org/abs/1810.01785