Update from RUH - I guess this will be relevant for some people.
The experience was very similar to when I self-transferred at CGK.

I arrived at T2 on an A3 ticket from ATH, connecting to a separate SV ticket to DXB from T4. No visa required, according to ExpertFlyer, an airside transfer would be possible.
I followed the signs to connecting flights.
There was a VoA desk and I asked about the rules for the transfer. She told me she doesn’t know the rules, she only issues arrival visas, and instructed me to wait. Someone would be picking me up.
Then nothing happened for a long time. I asked several times how long this would take, she couldn’t tell.
Finally, an impatient gentleman arrived.
I explained I was connecting on separate tickets. He talked to someone over the radio: “We have no transit passengers from
Athens!”
”Like I said, I have separate tickets…”
”Separate tickets, you take eVisa!”
”But I read that you don’t need a visa if your connection time is less than 12 hours and -“
”Separate tickets, eVisa! Go over there!”

I was then instructed by the lady to use one of the VoA kiosks, where she helped me skip most of the fields, just entering a single letter into them for me: “This is not important!”

I then had to pay SAR 395, about $105.
I tried to protest, repeating that this shouldn’t be necessary, and that I should at least be able to get a cheaper transit visa instead.
”The machine only issues tourist visas! But it’s a multi-entry visa, valid for one year. The transit visa you can only buy from the airline online when you buy your ticket!”

My fingerprints were taken twice, first by the kiosk, then by the lady at her desk. Then she stamped my passport and told me to go downstairs to the baggage claim area and through immigration. At immigration, I was allowed to use the fast track due to the eVisa.
Then I had to go upstairs, exit the terminal and walk over to T 3/4, which was quite a trek.
Finally, at T4, I could use the checkin kiosk to change my seat for free (which wasn’t possible online) and print my boarding pass, with my SAS# already on it.

The lounge was pretty underwhelming, but that didn’t matter as they started boarding about 15-20 minutes after I was airside.
It took me over 1.5h from landing until I was back airside, so if you’re planning something like this, make sure you have plenty of time.
I had 2:45h to connect and it was just enough time, they started boarding the A320 45 minutes before the scheduled departure time.

Also make sure to either book everything on a single ticket (and apply for a transit visa at the same time), or just buy the eVisa in advance, so you don’t have to sit around and wait for 30 minutes, just for someone to tell you there is no airside transfer… (Just like at CGK, I suspect it would have been possible, there was a security checkpoint - but it wasn’t staffed and making me go through immigration was obviously much more convenient for them.)