The whole point of how interrupt handling works is that it returns back to the same state of the program already in progress when finished. The abstraction is such that the interrupted program doesn't need to care.
Even in those "old" systems of single address spaces and no protection, you're constantly getting timer interrupts, interrupts for I/O, etc., which your application might not have installed its own handler for.
I agree - my main point is that an OS is ‘just a program’ as well
I suspect we’re both making a similar point in a roundabout way - the operating system is both another layer of abstraction on top of the Instruction Set, while also making the programming process for that chipset somewhat easier (providing software interrupts etc. at the expense of bare metal understanding).
My argument is loosely that modern (x64) assembly is not so much targeting hardware as it is programming into a software abstraction (the operating system).
Even in those "old" systems of single address spaces and no protection, you're constantly getting timer interrupts, interrupts for I/O, etc., which your application might not have installed its own handler for.