All successful OSes live without such systems: Android, ChromeOS, iOS, MacOS, Windows. When I install program I just need to know which OS versions are supported - that's all.
This is fine if you have one single »OS version number« to point to. Good luck getting all the Linux distributions to standardise. The systems you listed are at an advantage there because they get to control the complete platform.
And do note that on Windows many third-party applications (used to?) come with their own versions of important system libraries that they then put into the official library directory, possibly overwriting different custom versions of the same important system libraries that other applications put there earlier. Windows may be »successful« (i.e., popular) but certainly not due to its sophisticated and foolproof ways of handling libraries. OTOH, this is something that most of the mainstream Linux distributions get mostly right from an infrastructure POV (as I described for Debian).
But heyu, they are planning to do that. We'll see if it'll work or not.
Note that that proposal talks about third-party software for Ubuntu only. In that case the Ubuntu folks are in the fortunate position of being able to pretend they're Microsoft or Apple, but it doesn't buy us a lot for Linux packaging in general. (And of course the Ubuntu guys are notorious for doing their own thing.)
Also note that they don't propose to do away with the traditional way of packaging stuff, which will still be used for the »base system«, so it's not as if this was an attempt to make simple-minded packaging »work for the whole OS«.
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