Yes, I also imagined that the ES domain URL which AWS provides takes care of request handling for all the nodes under it.
This issue was happening quite rarely before. Then we made a change to an entity which resulted in a nested storage for two of its fields. This drastically reduced the mass indexing time for it. And the timeouts are mostly happening for this entity (timeouts are still rare, but still would like to avoid them).
Will update with the findings on resolution.