If I change both words of the phrase ("near" to "almost" and "miss" to "missed"), also changing both parts of speech and the overall phrase type in the process,

If I change both words of the phrase ("near" to "almost" and "miss" to "missed"), also changing both parts of speech and the overall phrase type in the process, that does shift its meaning, yes. I'm familiar with the claim that "near hit" ought to make more sense (and with the Carlin sketch that either originated or popularised it). But in practice, people use the phrase "near miss", not "near hit" (unless actively making the George Carlin joke), and they don't use "near miss" to mean what you claim ("almost missed, but didn't" i.e. hit). So this, again, isn't really a claim about what the phrase can mean, but about what you'd logically want it to mean. Such claims are a bit counterproductive at the best of times—language doesn't always follow compositional logic, and sometimes we just have to live with that—but here it's especially odd because the phrase means exactly what the component words say it does: a "near miss" (noun phrase) is a miss (noun) that is near (adjective).