I wouldn't say it's so much a matter of believably as it is a matter aesthetic flow.

The King Slime hat, for example, is silly and fun, but it doesn't exactly fit into the style of Eorzea. It stands out too much as an alien element in its design. You could certainly have the technology to build such a thing, but it just doesn't look like anything anyone would make here.
The strength of a world is gestalt. How much one can perceive it as a whole. That was one of the strengths of such games like X, XI, and XII. Their worlds where vast and varied, but the technology, structures, creatures, and style all felt congruent. You can't just throw everything at the wall and hope it sticks.

A big problem (aside form the slow speed) is in that a motorcycle represents a modern style that doesn't really mesh well with the more classical world we have. This world is far more World of Balance from FF6 than it is Gaia from FF7. You could hand wave it as technology from the Garleans or Allagans, but nothing about those suggest they would build anything like it as Garleans tech is mostly airships, walking mechs, and factories while the Allagans seem to have just teleported everywhere. And in the end, it still comes off as a forced appeal to cool beyond the aesthetic of the world. Shoehorned in as an awkward appeal to FF7 fans.
Wheeled technology here is limited to pulled carriages and carts of wood and steel. A jump to motorcycles with large rubber wheels seems a bit heavy when we don't even have cars, even though we have walking mechanical monstrosities and airships. Making a brighter lightbulb won't necessarily make a longer ladder.

Maybe if you replaced the wheels with mechanical feet? It sounds weird, but at least it would mesh better with what we got.