The thing is if your alliance is working right, you should receive 150 cards (if rare) or 1500 cards (if common) as well. The card floating lets you level out the RNG if it's done right... (i.e. I get too many of X, give them away, I'm short on Y, my team fills my gap)
I think it's actually giving more credit than the screen shows for common cards. Testing has shown it gives 1 for each common interactively, but actually gives 10 upon restarting the application. Strange bugs there.
Also, as you tier up with more trophies, you can request more cards and higher rarity.