If you instead use the older SET CONTEXT_INFO, you can capture that with extended events. But I am not going to say that this is a feasible solution.
Given that session_context can be a lot bigger than context_info, I can understand why they don't make it available for extended events. But at the same time, the need to capture who is the actual user is very apparent, so I think that Microsoft needs to find a solution. Maybe a flag to sp_set_session_context that you want to track this particular session key.
Anyway, the place to suggest improvements to SQL Server is https://feedback.azure.com/d365community/forum/04fe6ee0-3b25-ec11-b6e6-000d3a4f0da0.