I think the foundations of these two distinct states are an important thing to understand. This does not answer your question in it's entirety but it does play a role.

Put simply, the Saudi polity from its early days was aligned with Wahhabis, including the founder of the school/movement, Muhammad ibn Abdul al-Wahab, whose doctrine played a critical role in the spread of the House of Saud's power and influence in the interior of Arabia during the period of the 'First Saudi State' (1744-1818). Although Saudi power subsequently expanded and contracted during the 19th and early 20th Centuries, their eventual conquest of the Hejaz ejected the pro-British Hashemites from Arabia and Wahhabism spread throughout their domains, serving as a means of differentiating the House of Saud from pro-Western elites, legitimizing their rule in the eyes of many.

Kuwait by contrast emerged initially as a fishing and commercial centre in the 17th and 18th Centuries, subsequently benefitting from the pearl trade and eventually oil, and from the start enjoyed a more cosmopolitan environment than the provincial Arabian interior from which the Saudis emerged, with Persian, Indian, and European traders (including the East India Company) playing a prominent role in the early Kuwaiti economy. As the power of the ruling Bani Utbah clans and later the Emirs of Kuwait from the House of Sabah was from the start based on trade and trans-/international connections as opposed to an ultra-conservative religious revivalist movement, it is not surprising that Wahhabism has generally had a harder time finding support among the Kuwaitis. Additionally, the close connection between Wahhabism and the Saudis has not endeared the school/movement to other Arab elites. The Saudis fought a war over territory with Kuwait from 1919-1920, and imposed trade sanctions on them for more than a decade afterward, so one can understand why Kuwait might not welcome scholars and clerics from a movement associated with the Saudis.