If we know A is independent of B, why isn't P(A|B,C) = P(A|C) necessarily true? Let's say we know that A is independent of B, or mathematically:
$$P(A|B) = P(A)$$
Then how come we can't say the following is necessarily true:
$$P(A|B,C) = P(A|C)$$
If the outcome of B doesn't have an effect on the outcome of $A$, then why would the outcome of $B$ AND $C$ have an effect on $A$ that is different than the effect of the outcome of just $C$?
Is there perhaps a simple counterexample to illustrate this? For example, let's say $A$ and $B$ are the event of picking an ace of spades from a deck of cards (each has their own deck of cards).

Let's continue your example with two decks of cards, supposing that one card is chosen randomly from each deck.
You have proposed that $A$ is the event that the ace of spades is chosen from the first deck and that $B$ is the event that the ace of spades is chosen from the second deck.
Now let $C$ be the event that the two cards from the two decks are of different colors. Then
$$P(A|C) = P(A) = 1/52$$
but
$$P(A|B,C) = 0$$
because $A$, $B$ and $C$ cannot simultaneously be true.
Note that any two of $A$, $B$ and $C$ are independent but the three events are dependent when taken together.
