Control over Mecca and Medina legitimized the house of Osman's claim to rule the Islamic world. The sultans of the Ottoman Empire, starting with Murad I, styled themselves as the Caliph of Islam in addition to holding Imperial titles for this reason.

But control of the Holy City came later.

Selim I, the Grim brought Mecca and Medina into the Ottoman fold in the early decades of the 16th century. The Sharif of Mecca declared him as protector of the holy city, conferring yet more legitimacy upon the ruling dynasty throughout the Muslim world. It also served to strengthen the claim to being Caliph. This event coincided with a shift in the expansionary strategy for the Ottoman Empire too. Whereas Selim I's forefathers focused on expansion into Europe (the Balkans and south-eastern Europe) Selim and his immediate heirs focused on expanding into the Middle East and Africa, most notably Egypt.

So why only the west coast of the Arabian Peninsula? Likely due in part to trading concerns. There was and is a brisk trade in that region between India, the Middle East and Africa, but also to bolster their Islamic bona-fides and solidify their claim to the title of Caliph.

As for later on in Ottoman History...

Bedouin populations proved problematic for the Ottomans during the 'Long 19th Century.' In Egypt, Jordan and the Levant, Bedouin populations resisted Ottoman attempts to define and 'see' these restive groups. They were more stubborn than other populations, often cleverly resisting the adoption of an identity as 'Ottoman citizen' as opposed to, 'A member of this or that tribe.' They resisted settlement schemes, conscription mandates, census attempts. They were wary of land registration, which made their traditional way of life more difficult if not impossible.

So Leviathonlx's comment, though brief, is quite correct. Little of interest in the interior of the Arabian Peninsula from an Ottoman point of view. More restive bedouin's who will resist any effort to mold them into good Ottoman citizens, who won't stay in one place too long, won't pay taxes and contribute to the collective defense by sending sons to join in the army.