I know RUH (too) well but not JED, and I rarely fly SV.

That said, I agree with your interpretation of the TWOV regs (assuming you don't have an Israeli passport ). I have two concerns:

- the random KSA cluelessness. I've never met an unpleasant immigration official but quite a few bored/inept/unhelpful ones. I'm sure it will work out fine but you need to be prepared for lots of hassle and queueing. Remain positive and smiling !
- SV's poor punctuality. Especially worrying with two connections.

If there really is no other option, call your local SV office and ask to talk through the transit process in JED and RUH.