I get that it’s super compact and easy to carry around. I guess their compactness allows for easy manipulation and slights. Probably why coins are popular too. And I guess, maybe, that their numeric nature allows for “mathy” tricks that may not work with other objects.
What would magic look like today if playing cards didn’t exist?
Cards are versatile and lend themselves to a wide vareity of tricks. Some card tricks use sleight of hand, others use prepping the deck, and there are some mathy ones as well, as you alluded to. Cards are a good entre into magic; there are lots of good tricks that are easy to learn.
In the absence of cards, you’d still have coin magic, which mostly relies on sleight of hand.
Magicians, fortune tellers and con artists (until recently, the exact same thing) were probably the ones who invented cards in the first place. They’ve been using them since the Middle Ages.
If not for playing cards, there’d be a lot more disappearing elephants and Statues of Liberty and nobody needs that.
If you’re working a large room, you need one of those HUGE decks of cards, which makes for an interesting game of poker after the show.
I saw Penn and Teller live, and they did a card trick with cards made from Aluminum sheets, at least 5’ x 3’. They shuffled them with two forklifts, and, picked a card out of the deck by clamping a Vice-grip onto it and using the forklift to pull it up.
I was working at a theatre when they did a couple weeks of shows there. Super nice guys. They had that trick as part of the act and we had to reinforce the stage floor for it, we had trap doors and what-not. They also would not let anyone touch their props, outside of their own guys.
^^ That would be a REALLY interesting game of poker after the show.
P & T are nothing if not subtle. 
Naturally, it’s on Youtube…
It’s so simple. They have extra cards up their sleeves.
Remember once on Letterman, many years ago, when the big card trick reveal was on Teller’s eyeballs? 3 on one eyeball, club on the other? Special contact lenses. I still get woozy thinking about it.
Years ago I saw them on a talk show (Letterman, perhaps) where they did that. I was laughing at the ridiculousness of it all. It was just a simple card trick, made large so as to involve the forklifts and other equipment.
^ Abracapocus! I have vanished his post! At no time did the fingers leave my hand!
I’ll be here all week! Be sure to tip your waitresses!
Uneducated guess- Playing cards were designed and meant for playing games and the design lent itself to a simple function: reveal or hide its identity based on its orientation. Face up everyone knows. Face down nobody knows. Face an observer, known. Face away from an observer, unknown. From there it became the most basic demonstration of magic. Identify the card(s) without the proper orientation. Eg. From it’s orientation you can identify the card and I can not. Now watch me magically identify it.
Scleral shells. They originally thought of using contact lenses but it wouldn’t work, you couldn’t see the image.
The shells are way more hardcore; it’s an opaque prosthetic fitting over the eyeball.
SQUICK! Didn’t know what they were called; thanx. I’d rather have the roaches in the top hat.
Last P&T sidetrack: some TV NYE show buncha years ago, they grabbed a guy out of the crowd, locked him in some crate and kicked it off the pier into the water. Then they left. The entire segment took all of 45 seconds!
It would look like the nerds playing turns with figurines and dice like in Warhammer…
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Whaaat?!?
As a kid in the 1970’s I checked out a book about magic from the school library. I recall that the first section was tricks with cigarettes, often lit. Harpo does a few tricks with a lit cigarette, as an example of how common they were. When smoking became less common, perhaps cards replaced cigarettes as a way to do a trick.