Crash due to separate event handlers for IO events and plugin events for ClientSession

Found while tracking TS-4507 and original fix on that branch.

Cleaned up handling regular events at the same time as plugin events. The original code relied on the subclasses overriding handle_api_event to handle the regular events, but the handler only handled the TIMEOUT event. Changed that to augment the subclasses' main event handler to call out to state_api_callout in the event of the plugin events.