I believe you as an honest person, but also I do believe that you are wrong.
funding those terrorists groups are not in small amounts and they are done in at least tens of millions each. Also, transferring these big amounts of money with briefcases of cash is very wrong, you cannot transfer all the oney in one time, and one package, and since existing even one rat, can cause a big money to be stolen or the whole process be exposed, no one does that regularly to transfer money to overseas.
So, what they do, is using the charity, or cultural collaboration, or building projects cover to transfer this money in big amounts to overseas, then they can comfortably use this money in the destination country after dividing money into some smaller accounts.
Also, there are serious limits(10K$ in USA as an example) for money transfer from one source to another in a year, in which above this limit, your transactions would be fully investigated for money laundering and terrorism. I clearly know that Saudi has the same laws as well.
All of this process, means that no big money transfer can be done to terrorists groups without permission of your intelligence service, which makes sense, otherwise Saudi would have been a law-less country.
About alliance of these groups, they are multiple different ones and some of them are OK with Saudi government and some of them are not. Just consider the oppositions in Syria. Are all of them against Saudi government, now? of course not, only a portion of them,like ISIS, are so. BTW, even for the ones who oppose your government, they may be paid to oppose the bigger evil side using the concept of enemy of your enemy is your friend