What happens if I don't have an A2 or Tidus or Orlandeau or, really, any heavy hitter? Should I still wait for Ayaka or should I at least attempt to get him? Because he seems really damn useful.
You're in a rough spot because not only do you not have a big chain unit, you also don't have Tilith either. Chain units for the rest of the year (likely):
Orlandeau
Nyx
Lorraine
Because you joined too late to have Durandal, you'd have to TM Cecil (Excalibur) to make optimal use of Orlandeau, in addition to requiring other TMs I don't believe you have current access to, like Dual Wield. It really can't be overstated how important DW is, and right now, you have none. That's crippling.
Nyx and Lorraine come with native Dual Wield and Lorraine's TM allows you to bestow DW to another unit. Prishe can't really make use of the modifiers on the TM but she would benefit greatly from the DW component. You'd need to TM Nyx to make optimal use of him via his dagger; Lorraine should be more or less fine out of the box.
In your situation I'd probably recommend passing on the old man and going for Nyx or Lorraine when their banners drop (about a month and change away in all likelihood). As for Ayaka, I'm not sure.
If you blow your stack on Ayaka next week, you still have time to get some resources up before Lorraine, if not Nyx. If you somehow manage to secure Ayaka + chain unit, you're set. I guess I'd probably shoot for Ayaka and then switch to conservation mode.
It's probably a really good idea for you to spend this week going through the story and getting all the lapis you can.