Given a space occupied by a Tiny enemy, is a Small creature unable to move through it even though a Medium creature could?

PHB p. 191 states:

[Y]ou can move through a hostile creature's space only if the creature is at least two sizes larger or smaller than you.

Consider a tiny enemy, such as a crawling claw, MM p. 44, occupying a 5-foot space containing a doorway. Two PCs, a Small-sized gnome and a Medium-sized human, are adjacent to it. The human, on her turn, uses her movement to move through the doorway, even though the crawling claw is in that space. Even if the optional Overrun and Tumble rules (see DMG p. 272) are in play, she needn't use them, because she is two sizes larger than her enemy. She does, however, treat the crawling claw's space as difficult terrain.

Next it's the gnome's turn. The gnome isn't two sizes larger than the enemy, so he can't move through the doorway. Period.

Is that really correct?

Small creatures can't move through tiny enemy creatures' spaces

Yes, your reading is correct. A medium creature can move through a hostile tiny creature's space, but a small creature cannot (without using optional rules, like tumbling or overrunning [DMG, p. 272]).

In some cases, this makes more sense than others. Keep in mind that small humanoids are about half the size of medium humanoids, and about a quarter their weight (PHB, p. 121), making them roughly the size of a five year old human. Even though these humanoids can be extremely strong, they also are light enough that a relatively small amount of force could reposition them. Certain tiny creatures, like a full grown badger or a hawk (or a Crawling Claw which could carry things twice the weight of an an average halfling), could reasonably impede their movement.

On the other hand, "tiny" is a size category that applies to creatures of quite a variety of sizes. This is a necessity of the rules, since "tiny" is the smallest size category and will apply to anything which is smaller than "small." By way of example a cat, rat, and (normal) spider all all tiny sized creatures, even though these creatures outweigh each other by a factor of 20 (20 spiders to a rat, 20 rats to a cat). As such, a DM might make a ruling in the case of specific tiny creatures that ignores the usual rules on movement (such as deciding that a spider can't impede the movement of a small creature, but a hawk can). However, by the Rules as Written, a small creature cannot move through any tiny enemy creature's space.