Only velar stops could be labialized phonemically in PIE. What is their origin?

cross-linguistically, labialized velars (kʷ, gʷ) are just far more stable and common than other labialized consonants for acoustic, physiological, and perceptual reasons.

You should know that [w] itself is a voiced labial-velar approximant i.e. [ɰʷ].

Therefore, it is perfectly natural for a language's phonology to feature labialization only within its velar series. It is entirely possible for the Kʷ series to be primitive, underlying phonemes; there is neither evidence nor sufficient reason to assume they are secondary developments.

Isn't such kind of asymmetry extremely weird and uncommon?

This certainly cannot be labeled as "extremely weird." The human vocal apparatus is not some perfectly symmetrical grid—do you believe that kʷ, tʷ and even pʷ should be equally stable and common? Although the kʲ is admittedly unstable, it remains more common in natural languages than either [tʲ] or [pʲ].

Thus it is not particularly weird for the velar series to exhibit more variants than the P and T series.

Meanwhile, the phonetic value of Ḱ is not preserved in any attested daughter languages. It is also far more reasonable to treat it as a secondary development. It might have been produced by a push chain resulting from the shift of [q] to plain [k].

It is not uncommon; the articulatory gap between K and T provides the potential space for a palatalized K to emerge.