Gsrr

I pretty much agree with you, though personally I'd put assassin and rider over berserker for you. Reason: assassin second because you said you wouldn't mind getting Jeanne. I'm assuming you also wouldn't mind Jack since she's so good, so there's basically a 100% chance you'll be okay with what you get. Rider 75% chance.

Now, as for why I recommend berserker the least out of the four, is that, at least from personal experience using Vlad (NP2) he isn't amazing. He is quite mediocre unless you have the proper support for him, usually Tamamo. As for nightingale, she's okay as support but to me it looks like your priority for berserkers should be in damage, in which case she isn't great. This means that Cu alter is really the prize here, at a 33% chance to appear, less than the other 3 banners.

Of course, I have made quite a few assumptions here, so take all of this with a grain of salt and only where you agree with my assumptions.

Happy rolling.

Since the like button still isn't working, I'll go ahead and say I agree with zer0z. That said, Jack skills are kinda eh (imo) and getting a caster support (especially waver) would open up your friend support options.

If you don't have any of the 5* casters, your choice is easy. If you didn't already know, this game is centered heavily around the support casters. Tamamo is a requirement for Arts teams, while Waver is an all purpose God-tier support. Sanzang pales in comparison but that's mostly because the other 2 are absurdly good.
For Assassin + Extra, You have Shuten so I think you're fine. You don't really NEED Jeanne.
For Berserkers, Vlad needs Tamamo at minimum to be usable as his NP gain is shit. Nightingale is useful for debuffing and healing but will die quickly as she is a Berserker. Cu Alter is outstanding, and functions well independently, not to mention he is damn rare as he has only 2 solo rate ups so far in JP.
Riders are mostly excellent as well, but seriously, you would get more out of the Caster banner.
In summary, roll on Casters for gameplay and to make crafting teams easier. Roll Assassin + Extra for a solid assassin or your waifu. Roll Bersekers for a chance to get the rare Cu Alter. Roll for Riders if you don't want to roll on the other banners.

I think you're right about Caster being the #1. It's got pretty top tier servants and they're pretty vital to a team that you want to go far with (Gamepress thinks so too with all those high tier casters).
I'd avoid Archer because the archers in this Bag is sort of doo doo and you already have a decent Lancer.
Feel free to roll for Rider as well (Martha isn't bad but) because they have outstanding servants in the campaign (Except for Medb, sorry my queen :().
Assassin/Extra isn't bad too, Jack would be a great addition and Jeanne as well.

All in all, i think Caster would be your safest and most viable bet.