I'll answer at least one of your questions: at least for Mecca, this rule was not put in place by the House of Saud. During WWI, both France and the UK were in close contact with the Hashemite Sharif of Mecca, and Mecca was closed off to non-Muslims, giving France an advantage because they had Muslim officers of a relatively high rank, which the UK lacked.

Source: Rémi Kauffer, Histoire des Hachémites.

This rule goes back a long way and predates WW1. It is useful to consider the account of Sir Richard Burton who published an account of his travels in "Personal narrative of a pilgrimage to Mecca and Medina" published in 1874. Due to its age and popularity, the book has been OCRed and the full text available for download.

Sir Richard Burton has to assume the disguise of a Sufi with Indian origins to make his visit in 1855-56. He was an excellent linguist and had spent a lot of time travelling in India and Arabia so he could get away with it.

However, Burton was not the first. The place had been visited several times over the years back through 1504 by Ludovico Verthema, but the problem of having to visit in disguise didn't seem to happen until the early 19th century with the visit of Catalan Domingo Badia y Leblich when as a result of the political and religious problems, the cities became closed in effect even not in fact (i.e., you would probably be murdered or arrested as a spy).

Apart from Burton, mention should be made of John Lewis Burkhardt a Swiss who also travelled there and slightly predated Burton. He wrote the two volume book "Travels in Arabia" which gives a good account of the Haj pilgrimage.

A good summary of the general travel history is "Literary History of Travel and Exploration: A-F" Edited by Jennifer Speake page 32 in "Arabia".

Your question has already been sufficiently answered above but I just thought I'd mention an interesting fact about the French assistance to the Saudis during the Grand Mosque Seizure in Mecca in 1979. The Mosque has been seized by a few hundred very well armed fundamentalist jihadis and the Saudi army wasn't up to the job of taking it back. They (secretly) had to call in the French special forces to help and so just before they joined the attack a number of French soldiers converted to Islam on the spot. Source: Trofimov, The Siege of Mecca