Stealth Audio Player can do exactly that. There is “Activate for ALL APPS” option.

You can install alternative app to handle sharing. Sharedr for example.

But it won’t work on android 12+
(unless you remove system intent resolver)

Based on GrapheneOS’s response in this thread, I understand that freezing apps might be completely unnecessary

Link in that part doesn’t work but is it really that bothersome to use Hail with Shizuku? You only need to re-enable shizuku after reboot. I don’t restart my phone too often so it is not a problem for me.

In my case I rooted my device once to try some fancy things. It was fun but then I never bothered with rooting again. Most things that I need can be done with Shizuku/ADB and/or app patches.

If you really need it not to be killed, you can always open split screen before opening the second app, or just open it in free-form mode. But this is more of a workaround than solution.