If Saddam invades Kuwait as OTL and the US/NATO only support logistically game over. A question is what does logistic support consist of - free or cut rate munitions, aircraft, tanks, or only what the anti-Saddam forces can pay for at market rates? Will the US/NATO draw down war stocks, transfer supplies from POMCUS in Europe etc as was done OTL because production rates would need to be ramped up to cover the supply if not, and that takes time in any case.
For the Saudis, and the other Gulf States, a very high percentage of maintenance on aircraft, tanks, etc was done by contractors. OTL under the umbrella of US/NATO protection they pretty much stayed put, without this a fair percentage will decide that having SCUDs coming down or even "old fashioned" air raids is not what they signed on for. This will cripple the Gulf Arab forces severely, and while the Egyptians or Syrians might do better, still an issue. (1, 2) Will the US/NATO have E-3s managing the air battle, if not the anti-Iraq forces are going to losr a biog advantage they had in the air war.
Even if the Saudis (and others) can recruit LARGE mercenary formations it will take time for them to arrive, get sorted out, and get logistics in place. Unlike the US (and to an an extent NATO) forces they may not have as much commonality with the logistics already in place in the KSA. Several thousand South Africans or Eastern Europeans are fine for a campaign against some tin pot dictator in a nowhere country, but for all its faults Iraq had a substantial military with decent weapons and a modicum of organization.
A final issue is that of geography. The distance from the KSA-Kuwait border to the causeway to Bahrain and the oil fields of eastern KSA is not really all that great and its pretty flat desert/scrub. Frankly had the Iraqis planned to keep going they could have gotten there even before the US could have sent more than part of the 82nd Airborne with limited supplies. Against the KSA military plus some Arab helpers from the Gulf States who get get there quickly...
(1) In the US military, tank crews can do a good deal to deal with mechanical failures or minor damage to their tanks. During the first Gulf War I heard stories over there how if a Saudi tank broke down, the crew made no attempt to fix it but immediately called for a recovery vehicle to take it to a maintenance area.
(2) The port of Al-Jubail, about 80 miles from the border, had a very nice modern hospital. Most of the medical staff were "Europeans" (specialist doctors and supervisory nurses etc). When Iraq overran Kuwait and it was unclear if they were stopping, all of these folks decided it was time to leave and the hospital stood almost empty with only minimal staff. An example of how dependent the Saudis and other Gulf States were on "Europeans" in many areas.