Which sounds better 2 way or 3 way speaker design

Three way speakers with concentric mid/tweeter designs are another twist that offer very good results. 

6 of one. The square root of 36 of the other.

There can be number of variables approaching infinity related to speaker parameters and considerations. I’m with @skiznfliz on this one: put them in your room and listen to them.

To say, as the OP does, that "a 3 way design will give the listener a much better chance to hear the full audio spectrum as opposed to a 2 way design" is like saying a car with five speeds must go faster than one with only four. It reflects no understanding of what drivers in a speaker actually do.

General rule of thumb...Drivers handling approx 30 hz to 300 hz should always be NOT the same driver that handled 300 to 2000 hz.

Having said that, a guy who put 2 dollar trash drivers in a 3 way and designed a trash crossover/ cabinet can make a 3 way sound worse than a guy who put more money and competence into a 2 way.

Parts and build can make a huge difference, but so can how they’re used. For instance, I have a pair of old Yamaha 2 ways speakers than I’m using in a desktop system where they sit 2.5’ from my ears that sound great!  But there is no way I would put them in my main system and expect them to sound as good.