Maybe this is a problem of semantics, or something that I simply can't quite understand any longer because I am an electrical engineer (or at least I have a diploma that says I am :-) ). But this distinction is very strange, and I'm not sure why/how you're making it. Electronic engineering is nothing like that.
Knowing rules of thumbs and how to "combine components" into something useful is not a parallel skill set to electronic engineering, it's a part of electronic engineering. Rules of thumb, models and reference schematics (from manufacturers or whatever) are pretty much how you come up with the first draft of a circuit.
Also, electronic engineering consists precisely of designing circuits. Quantitative analysis is a big part of it because without it you can't always know whether the circuit you've designed works according to specs (and in practice, yes, experimentation and testing plays a big role, but there are only so many prototypes that you can blow up before you run out of time, and only so many things that you can test for in a regular lab). The whole point of one's activity as an engineer is to combine components into something useful.
Sure, you don't sit in a lab drawing schematics and building things all day, because there are a lot of other activities that go into building a good product. There's a lot of validation work and a lot of analysis and a lot of planning. Some engineers focus their time only on one of these things, especially because they really are so complex, and so complicated, that you can spend a lifetime studying just one of them and you're still left with a lot of stuff to know.
But at the end of the day, designing electronic gizmos is pretty much what you do, regardless of what role you're playing there.
You can certainly make things by sticking components and circuits together without a thorough understanding how they work. But the idea that engineering somehow mostly about understanding a system through mathematical modelling before building it, and design is mostly about making things by sticking components together by intuition, is very much absurd.
In my time this was basically a distinction between an engineer and a technician - technicians generally having a lot more practical experience, but lacking in the theory, while fresh out of Uni engineers knowing the theory well, but lacking practical skills in solving the real-world problems. With time a good engineer picks up those skills too.
Good summary. I'd also add that 80% of the difficulty is spent on optimizing the last 20% of performance (i.e. the last dB), which is what separates the hobbyist from the pros, or the art from the quantitative.
Knowing rules of thumbs and how to "combine components" into something useful is not a parallel skill set to electronic engineering, it's a part of electronic engineering. Rules of thumb, models and reference schematics (from manufacturers or whatever) are pretty much how you come up with the first draft of a circuit.
Also, electronic engineering consists precisely of designing circuits. Quantitative analysis is a big part of it because without it you can't always know whether the circuit you've designed works according to specs (and in practice, yes, experimentation and testing plays a big role, but there are only so many prototypes that you can blow up before you run out of time, and only so many things that you can test for in a regular lab). The whole point of one's activity as an engineer is to combine components into something useful.
Sure, you don't sit in a lab drawing schematics and building things all day, because there are a lot of other activities that go into building a good product. There's a lot of validation work and a lot of analysis and a lot of planning. Some engineers focus their time only on one of these things, especially because they really are so complex, and so complicated, that you can spend a lifetime studying just one of them and you're still left with a lot of stuff to know.
But at the end of the day, designing electronic gizmos is pretty much what you do, regardless of what role you're playing there.
You can certainly make things by sticking components and circuits together without a thorough understanding how they work. But the idea that engineering somehow mostly about understanding a system through mathematical modelling before building it, and design is mostly about making things by sticking components together by intuition, is very much absurd.