It seems like a possible oversight but the Base is only added to random scenarios 7. Vanquish The Villains and 8. Solve The Mystery. The original 6 from Dragon's Demand haven't been updated to include the base (unless I have missed something)
I don't think the original six need the Base. The Base is a CotCT thing. Perhaps if we'd gotten random scenarios 9 and 10 in the next AP, they'd feature something unique to that AP.
Honestly, I wish the Base wasn't listed for scenarios 7 & 8. I mostly use The Wandering Way as a way to add a "side quest" to campaigns, perhaps something to play if one player can't make it that week and we don't want to get ahead of them in our campaign. To have to use the Base for a scenario I'm skinning as some very non-Korvosan setting feels foolish. I'd have preferred a sidebar saying that if you're playing a random scenario with characters who are in a campaign that uses supporters, include the Base. (Those are the only characters for whom the Base will be any good anyway. Otherwise, it will just close the moment no players are there.)
I hadn't considered using Random Scenario for larger groups to tide players over.
Currently I am playing in a group of 3 and we are always all present for every mission, and only just now in CoCT #3 decided to try the random to try to get a few more attempts to get Lv.1-3 equipment we like before taking on the Lv.4 challenges.
I'll have to consider that for when we play the DLC campaigns with a larger group.
I initially asked about the base because I'm greedy and want the "draw new x" powers to loot more items per mission but how last night went (each character got 1-2 new toys on average when we played Secure The Area without the Base) I think the game still feels rewarding R.A.W. (though that was partly due to our Kyra drawing a random Restorative Touch out of the box as her scenario reward)