Doing this without any loss of functionality would likely require replacing Amazon's backend servers.
Which is unlikely but even if done, you would still be sharing data with someone else.
The most reasonable alternative is to replace the hardware. I use Amcrest which supports RTSP. But then you'll need software to monitor and record. I use Blue Iris. But then you'll probably want a way to access your PC remotely to check on things. I use MS RDP but it needs custom setup for reasonable security (custom port, timeout for too many invalid logins, port forwarding on your router, etc.).
All things considered, this ain't happening for most users and I'm pretty sure Amazon knows it.