Hi Oliver,
According to your description, you think the native mode of SQL Server Reporting Services will be deprecated. Right?
SQL Server Reporting Services is supported in Chrome on windows operating system. See:
Planning for Reporting Services and Power View Browser Support (Reporting Services 2014)
If you have any displaying issue in Chrome, you can install IE tab for Chrome to access report manager.
If you still have problems, I would recommend to submit a feedback to the Microsoft Connect at this link
https://connect.microsoft.com/SQLServer/Feedback. This connect site will serve as a connecting point between you and Microsoft, and ultimately the large community for you and Microsoft to interact
with. Your feedback enables Microsoft to offer the best software and deliver superior services, meanwhile you can learn more about and contribute to the exciting projects on Microsoft Connect.
If you have any question, please feel free to ask.
Regards,
Shrek Li