At risk of sounding like Ron Swanson... a wheel is a structure. Build it wrong and it will fail. Build it right and it will serve you well. That's called engineering. Engineering is a science.
Whilst I agree that engineering is a science (in fact it's all sciences in one discipline), really really good engineering is hard to differentiate from art. It's about doing something so unfathomable to the normal person by applying principles and knowledge that often isn't understood by those outside of your field of engineering that those around you look, interpret, but do not fully understand.
That's art, the Falkirk wheel, Strandbeest, ships that sail the world, tiny electrical circuits, process systems they are all engineering, but to me, they are art, they are things that can be stood and looked at with absolutely no understanding of how it works, no comprehension of the materials used, just amazing feats of engineering that transform the world around us in to a better place.
A good wheel builder, yes what he is doing is engineering, applying skills and knowledge to a set of materials in order to build something that it isn't understood by anyone outside of that discipline. something that if done well will is often proclaimed as "stunning", yer, that's art. it's not a dark art, it can be learned, even easier to learn these days with interwebs. todays tools remove some of the skill from it (spoke tension gauges for example that allow anybody to balance tension) but that's not the point, it's still a skill.
art is an extension of engineering has been since the days of Da vinci, we as a culture have separated it.
Parting comment, wizards and witches were prescribed the term "dark art" in the middle ages. they weren't magic, they just had more knowledge and skills than the normal person. you go back now and you'd be branded a witch and burned at the stake. this is where the term dark art came from and it's fairly obvious why it's been applied to wheel building over the years.
I'm not arguing with you, I agree it's engineering, I'm saying that to people without the knowledge, even simple engineering is a dark art they simply don't understand.