The idea is that software would only be shipped if it actually works, and part of this would be ensuring it compiles. I'm not suggesting we use Make files or some other foul approach like that. Code should ideally be distributed as a single zip file with an executable header that triggers it to be unpacked in some cache directory by a program analogous to the linker, and compiled by the simple process of compiling all source files. Additionally this could be sped up drastically by having the compiler be a single executable with uniform flags across files. Header files could be parsed once and shared between build threads in parallel, which alone would safe much work on the part of the compiler. If this still proves insufficient, a language with superior semantics to C could be chosen which can be compiled faster. Most code these days shipped is JS which is compiled on the user's computer. This would be an order of magnitude improvement on that from a performance perspective.

No, end users need not get involved, it could/should be handled by the operating system.

Just because you are used to slow compilers doesn't mean fast ones are impossible. As I said in the original post, code optimisation can be done before compilation on the developers machine, so all that need be done on the target is a simple debug build. An example is Jonathon Blow's JAI compiler which compiles around 250,000 lines of code per second. Even on very slow hardware this is reduced to perhaps 80,000LoC/s, two minutes for 10 million lines of code. Most users would be tolerant of an "install" box with a progress bar that takes at most two minutes for only the largest application on the computer. Or perhaps the browser people would opt not to distribute their software in this way. Advanced software may have advanced needs, beyond that of the average application. Most code is already written in Javascript these days, meaning it begins life being interpreted and is JIT compiled on the user's computer. Few seem to notice the compilation but many complain about the general sluggishness.

If you offered users a deal: spend 1 minute installing the software and it is snappy and responsive when you use it, I suspect all would accept.

This is somewhat exaggerated.

The compilation of firefox could take a few hours on some laptop dual-core Skylake CPU from 10 years ago.

Nowadays, on any decent dektop CPU with many cores the compilation of Firefox should take significantly less than an hour, though it remains one of the handful of open-source applications with a really long and non-negligible compilation time.

The Linux kernel is normally compiled much faster than Firefox, except when one would enable the compilation of all existing kernel modules, for all the hardware that could be supported by Linux, even if almost all of that is not present and it would never be present on the target computer system.

Slow and usually pointless for minor updates.