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).

I quote Joseph Blitzstein's Introduction to Probability (2019 2 edn), p. 65.

Example 2.5.11 (Independence doesn't imply conditional independence).
My
friends Alice and Bob are the only two people who ever call me on the phone. Each
day, they decide independently whether to call me that day. Let A be the event
that Alice calls me next Friday and B be the event that Bob calls me next Friday.
Assume A and B are unconditionally independent with P(A) > 0 and P(B) > 0.
However, given that I receive exactly one call next Friday, A and B are no longer
independent: the call is from Alice if and only if it is not from Bob. In other words,
letting C be the event that I receive exactly one call next Friday, $P(B|C) > 0$ while
$P(B|A,C) = 0$, so A and B are not conditionally independent given C. 

