I think this idiomatic logic is totally bogus. Because, literally speaking, a third wheel is not terribly awkward. A fifth wheel, however, is. Consider the following:
If you have one wheel, you have a unicycle. Unicycles are rad. Anyone who can ride one is worthy of your respect and mine. Nothing awkward there.Vehicles don't get awkward until they have five wheels. Because a three-wheel vehicle isn't awkward, but a five-wheel vehicle is, I therefore declare "the third wheel" to be a stupid and illogical idiom. Henceforth, it is to be replaced with "the fifth wheel," even when there's only three people.
Add another wheel, and you get a bicycle. This is a fairly standard and extremely functional mode of transportation. Add a motor and you have a motorcycle. Add a leather jacket, and you're cool. (Add gelled hair, and you're the Fonz.) Fifties stereotypes notwithstanding, there's nothing awkward about bicycles
Add another wheel, and you've got a tricycle. While often associated with the younger generation (read: toddlers), the tricycle is still an extremely usable means of getting around. Add a motor, and you get a pretty sweet motor-trike. This isn't really awkward either.
One more wheels gets you a car. Alternately, an ATV. Either way, it's not awkward.
Add a fifth wheel. Where does it go? What does it do? How does it steer? This is when things get awkward.
A note: Some might try to make the case that the fifth wheel is a spare tire on a car, and thus not awkward. This excuse, while creative, doesn't quite work. Unlike the rest of the car's wheel, the spare tire does nothing. It just sits there. That in and of itself is awkward for the other four wheels.
Additionally, if you carry the analogy further, a while that does nothing must be analogous to a person that does nothing. For a person to do nothing, they'd have to be asleep or unconscious or something. (Depending on how this person got to be asleep or unconscious, they may or may not be awkward for the other people.) So if two people are on a date, and they have a third person asleep in their back seat/unconscious in their trunk, then the third guy is a spare tire.