They don't call him Conjob for no reason!
You're absolutely spot on; John was never the traditional magician. At least not in the sense that he'd shoot spells from his palms (well, until he did...). What made Constantine stand out isn't that he was proficient in magic or had any capabilities; he was just some guy that picked up on occult knowledge and went forward with it.
Even more, the one thing that is special about John is his demon blood and that was used frequently as a negative power. It seemed like an outright reversal of the normal superhero idea where a character gets to have abilities because of blood or a serum or something like that.
Ennis was quite good at separating John from actual spell casting. A lot of what he did involved sigil creation and empowering rituals. The Shows have taken liberties with that, making Constantine a slight bit closer to the magical 52 version, but there's this moment at the end of the first episode of Constantine: John's got flaming hands, but instead of just a spell, they have him break up a lighter, rub the fluid over his palms, and then light himself up. Protection magic, but still a rough, almost sleight of hand kind.
I'd say the direction it went made Constantine more appealing to a broader audience, but I don't even think that's true since the more human he is, the more people seem to like him.