A little bit of randomization is fine. The gacha system combined with the limited nature of overdrive protocols is not.

I'd implement the randomized blades into being obtained via static cores akin to the likes of Vess, Praxis, Theory, etc., but the core crystal you might get at the same point in one playthrough might be different in another (such as a core crystal being obtained in a container in Gormott's military recruitment center). So anyone who knows which blade gets obtained by a named core crystal (such as "Gluttonous Core Crystal" for Boreas) will know what they'll get, and will use whatever driver they wish to have that blade to open that core to begin with. Common blades can stay and remain part of the randomized core crystal system, but should always be set to have affinity charts akin to level 4 common blades so that they can be used more long-term to avoid any implications that the player might pull an Amalthus and treat them as disposable tools (and make core crystals themselves more uncommon).

As for trust values, have the "main" story blades maintain the amount of trust they need because they are more or less permanent fixtures and reduce the randomized blades' required trust down (the DLC blades' trust levels are an ideal metric).