Having no relationship with Cuba for decades did wonders for us and Cuba. Now that we have relations, they still are run by the same people.
How about all those years we refused to even recognize that China existed. (We ended up fighting Chinese soldiers in the Korean War.)
We do not have to like or agree with foreign nations to maintain good relationships with them. Throughout the cold war, we maintained a relationship with the Soviet Union, while spying on each other, and fighting each other h trough proxy wars. Even when they threatened to bury us, we maintained relationships with them.
The argument against maintaining relations with Saudi Arabia or Iran or anyone is illogical and counterproductive.
There are many problems faced by the earth that cannot be solved by the US working alone. Global warming, for one, will need the cooperation of all nations. If we do not have relationships with a nation, we can not begin to have a dialog about solving world problems. We don't have to like their form of government, but we do need to have some form of understanding and long term relationship.
Your analogy is false. The question is not whether we have to have the House of Saud as President or adopt their customs and laws. The question is whether we an have a constructive relationship with their religious government, whether we like it or not.