Inter-satellite links simply provide additional (time-variant) paths, which doesn't inherently relate to shared state.

You seem to be under the impression that inter-satellite links somehow imply a self-organizing mesh topology that preserves terminal-to-gateway associations at any cost (including that of extra in-space hops), but that does not necessarily follow from the existence of ISLs.

In other words, your observation of occasional routing instability causing higher-layer issues is perfectly compatible with working ISLs.