"Unrestricted access to airspace" is a misleading headline.
An Open Skies agreement means that Saudi carriers can operate scheduled service from Saudi Arabia to any US airport, and US carriers can operate scheduled service from the USA to any Saudi airport. There may be small print allowing for intermediate stops or "beyond" rights, but basically that's all it is - a liberal bilateral air services agreement. This agreement itself has nothing to do with either the US or the Saudi visa regime, with ease of access, propensity to sever limbs, etc.