According to Richard Bulliet's History of the World lectures (available online), four-wheeled wagons are globally less common than wheelbarrows or two-wheeled carts because:
(a) extra wheels add considerable weight and friction
(b) wagons have a larger turn radius (See also https://en.wikipedia.org/wiki/Wagon_wheel_(transportation)#T...)
TFA answers the question "why a wheelbarrow and not a two-wheeled cart": "Compared to a two-wheeled cart or a four-wheeled wagon, a wheelbarrow was much cheaper to build because wheel construction was a labour-intensive job. Although the wheelbarrow required a road, a very narrow path (about as wide as the wheel) sufficed, and it could be bumpy."
It's true that the eastern and western wheelbarrows were different, but the Chinese also used "western"-style wheelbarrows for moving bulk materials short distances on construction sites, whereas we had no equivalent of their wheelbarrow for longer distance travel. Sounds like they won there.
As for adding more wheels, why? You're just adding more weight. Keep in mind that these need to be pushable by a single person. A single large wheel is the best design for handling rough terrain and optimizing maneuverability, while still being pushable by one person. By the time you've added multiple wheels you're talking about a larger type of vehicle that needs better roads and draft animals to pull it.
The article says: "Compared to a two-wheeled cart or a four-wheeled wagon, a wheelbarrow was much cheaper to build because wheel construction was a labour-intensive job."
The article is in a way very misleading.
'Wheelbarrow' in 'The West' is generally a local tool, used to move things like dirt and what not 50 feet to somewhere else.
What the Chinese have there is a '1 wheeled cart' for transportation, which in the west would have '2 wheels' or even '4 wheels'.
And yes, one would be right to question why on earthy they wouldn't add another wheel or two. Maybe resources?