The US-Saudi relationship played a part in that it was rare that a Saudi National was refused entrance into the county, and if that person over-staid their visa there usually was no penalty. But when you consider how many Saudis there are, and how many of them were Al Qaeda 9-11 hijackers it's ridiculous to say we could have drawn a correlation in advance.
I mean, it wasn't exactly a secret to US intelligence that Saudi Arabia has not historically been helpful about combating al-Qaeda, especially prior to the 2003 bombings in the country - and even then, the Saudi regime
still seems to have a lot more tolerance and possibly covert support of some of these militant groups
as long as they operate outside Saudi Arabia and aren't mounting attacks within the Kingdom.
Osama bin Laden's high school science teacher Ahmed Badeeb was one of longtime Saudi intelligence chief Prince Turki al-Faisal's top aides. He has openly said "I loved Osama" and "He was our man" in Afghanistan during the 1980s.
When considering just how much Saudi funds (including the extended bin Laden family's) were channeled into the "Arab Afghans" and the most stringently reactionary-Islamist of the Afghan mujaheddin from intelligence sources, from charities, from religious fundraisers, from prominent business families (again, like the bin Ladens), and from the personal coffers of Saudi royals and senior officials, and when considering how popular Osama was, how admired he was for his personal religiosity and for giving up a cushy lifestyle in Saudi Arabia for the war in Afghanistan, and all of the strong relationships with important Saudis (and important players from across the Islamic world, but especially in the Gulf monarchies and obviously Pakistan and Afghanistan),
it simply strains belief that he didn't maintain at least some of these important contacts even after he was officially disowned by the Saudi regime along with his own family (a rather slow process that stretched out multiple years in the 1990s, by the way).
This puts alleged or real Saudi support for bin Laden/AQ in some context. Honestly, I think the "exporting" of Saudi jihadists or would-be jihadists was as much about hoping/praying that they would be martyred on a battlefield in Afghanistan, Chechnya, or Bosnia as much as anything else. Dead suicide bombers in a war zone aren't a threat to regime stability, after all.
And remember that the Soviet invasion of Afghanistan happened right as the Iranian Revolution was in full swing and Khomeini was consolidating his power. The Ayatollahs were saying Death to Israel, Death to America - and Death to Monarchs (like the deposed and exiled Shah; none of this was lost on the House of Saud). Saudi Shia were rising up in the Eastern Provinces (where the oil is, incidentally) at the same time as some Muslim Brotherhood-influenced Sunni fanatics had seized the Grand Mosque in Mecca, provoking a bloody fiasco of a siege that left many hundreds of Muslims dead.
All of this heavily religious discontent and instability had to be dealt with somehow, so the Saudis did what they saw as rational: get all the jihadist crazies into Afghanistan where they could martyr themselves while granting far more power and influence to the Wahhabist clerics at home.
And note that this was all before a victorious and long-supported by his family and the royals who sponsored them (and him) Osama bin Laden came home and offered his services as a Saudi patriot and a pious Muslim fundraiser and organizer of jihad to expel Saddam from Kuwait - and was promptly snubbed
in favor of the Americans, who proceeded to station hundreds of thousands of soldiers in the Land of the Two Mosques -
Osama's land - with the formal blessing of the same Saudi royals and state clerics who refused his offer of help against Saddam. Needless to say, the rest is (horrible and tragic) history.