Basically I agree with your hypothetical, though it wasn’t exactly the point I was making, but very close.
First, yes, many people can hear something like that solo play it back on the guitar, or figure it out in not that much time, and not be able to tell you the intervals, roman numerals for the chord progression, note names, etc.
And many people out there could tell you the intervals, roman numerals, etc, but perhaps struggle to do it by ear very quickly.
I think that people general refer to music theory when they’re talking about the practice of naming things; naming conventions, naming common occurrences. We call a D minor triad a D minor triad instead of a saying “Well, it’s this note, that note, and that note” because the notes are used together often enough that it’s convenient to name them something. Same is true for something like a minor plagal cadence or modal mixture or anything simple or complicated.
But back to the minor triad as an example, yes, some people can play a minor triad when they hear a minor triad yet not know (or care) it’s called a minor triad. However when those folks then learn that it’s called a minor triad, if they remember that it gives them means to organize the sound like @jllopez articulated well. When you have the sound of a minor triad memorized, every time you hear it you know exactly what it is…then it becomes easier to build off of that, like memorizing the sound of different inversions, or a m7 chord, or a m6 chord.
If music theory is just naming things, then its isolated use is to look at music on a page and write an analysis of what’s happening. That’s pretty cool and all, but for everybody except music theory teachers, the goal is to use theory in conjunction with other skills to accomplish useful musical tasks like learning a solo quickly, knowing how to improvise over a chord progression, being able to choose from many options for what to do in the next measure of one’s own composition, etc.
Think of the music theory a bit like delegating to an assistant…it makes a lot processes faster and more efficient, but you need to tell the assistant what to do.
So when somebody works on their ears, they can do so without any organizational system, but it’s certainly a lot harder and less practical.
For whatever it’s worth, this skill as it applies to the Cars solo is a lot more common than you might think. It’s great that it got you excited, it’s just also a relatively simple bit of music for someone who has been spending hours every day playing and figuring stuff out, theory or not, for a few decades. It IS a notable accomplishment or benchmark, it’s also just not super rare.
I suppose there’s an encouraging message in all of this, which is, that type of skill is achievable for many! Just takes putting in the work on the ear training and other related stuff.
I have to figure out music for students all the time, and I also just simply enjoy transcribing, so I’ve gotten faster and faster with it or over the 25 years or so. Just like anything else, the more you do it…!