your weapons deal +1 damage […]
(Arcane Strike description)
Note that this is +1 damage—not a +1 enhancement on the weapon as a whole, as offered by, say, the magic weapon spell.
[…] and are treated as magic for the purpose of overcoming damage reduction.
(Arcane Strike description, continuation from above)
And again note “are treated as,” not “actually become.”
So Arcane Strike makes a weapon—including a card thrown using Deadly Dealer—deal extra damage, and overcome DR/magic. It does not give any bonus to attack rolls, and does not make the weapon count as magical for any other purpose.
Therefore, the +1 enhancement bonus to attack rolls that you get from a masterwork weapon—including a harrow card thrown using Deadly Dealer—is the only attack bonus on offer here. It isn’t “subsumed” by anything from Arcane Strike, because Arcane Strike only offers a damage bonus and not an attack bonus. Moreover, the damage bonus from Arcane Strike is untyped, rather than an enhancement bonus, and therefore stacks with actual enhancement bonuses—you could use Arcane Strike to increase the damage of a +5 longsword without difficulty.
Anyway, all that said, yes, Deadly Dealer is a very weak feat. You could very easily do better by just buying actual darts, to say nothing of the possibility of buying weapons that are superior to darts. The only way it could possibly make sense is if you already had a very good reason to have a deck of cards in hand during combat anyway—and since using this feat ruins a harrow deck, that reason can’t be anything to do with one of those. Even if you have such a reason, you’d further have to have a very good reason to feel the need to make mediocre ranged attacks despite whatever other things you’re doing with the deck in your hand, and then this need would have to be so pressing as to justify burning an entire feat, or possibly two, on it. I cannot think of any possible combination of requirements that could justify that.
For a wizard (or a monk), it’s that much worse—because, as you say, you need dart proficiency, and these two classes (and only these two classes) don’t have it automatically. That said, the Heirloom Weapon trait can get it without using a feat (though considering how powerful some traits are this is still a significant cost), or an opalescent white pyramid ioun stone could get it for 10,000 gp (again, that is quite a price, particularly compared to the 1,500-gp cracked version).
For that matter, Arcane Strike isn’t very good either—the damage bonus is small, and there is no attack bonus, so you don’t really get very much for your feat slot this way. Maybe a low-level character who has to worry about DR/magic but cannot, for some reason, get a magical weapon might want it, but that is a painful choice forced upon them by circumstances (substandard magical item access, or a particularly awkward fighting style like improvised or thrown weapons). Even then, there are much better, cheaper solutions, such as the aforementioned magic weapon which costs less and does more.