An "embedded system" can be as simple as "when this button is pressed for 3-5 seconds, turn on this light for 30 seconds." That hardly requires much of a skillset.
And as far as having all the things you mention in "one head," every member of my team can handle that easily and we're not particularly noteworthy!
In my own experience moving from pure applications software development (though not a web developer) to network appliances to lower-level embedded development i have been amazed at all the "hidden complexity" which you suddenly become exposed to. And this is just the software part. If you get into the hardware part you have a whole another world of knowledge to learn. Merely doing something without understanding is the difference between a "Technician" vs "Engineer/Scientist".