I’m not a whiz with the math involved, but I am of the opinion that 500 consecutive same-side flips is a large enough sample size to calculate that the coin in question is biased, so it would be unreasonable to assume that the next flip is 50/50.
https://en.wikipedia.org/wiki/Checking_whether_a_coin_is_fai...
The coin can be assumed to be fair before the flips, but after the flips we have no mathematical reason to believe that the coin is fair, as our results heavily suggest that it quite simply isn’t fair.
However, for the purposes of discussion, assuming an ideal coin, the probability of 500 same flips in a row is so statistically unlikely that fairness of the flipping process and/or flipper must then be called into question.
Even if the coin, flipping process, and flipper are all ideal (which wasn’t stated, but we will also assume for the sake of argument that we have ideal immovable goalposts), the likelihood of the 500-in-row event is so improbable as to be unreasonable to use as an example or even a metaphor, because it doesn’t have much predictive power in a conversation, as ideal coins don’t exist any more than ideal coin flippers.
Even a coin designed to be fair would be fairly deemed defective after so many same-side flips, and any reasonable gambler would demand that the coin be replaced with another; at that point an argument could reasonably be made to also change the coin flipper and/or the venue.
I agree: the events are independent in probability, which was never under debate, and is a separate matter to whether or not the coin is fair.
As for your edit:
> Pretend I said 5 instead of 500 if it makes you feel better.
At this point, wouldn’t you say that’s a bridge too far? It’s uncharitable to ask of your interlocutor as that changes the entire context and implications of what you originally said upthread.
Asking me to just pretend that you made a different argument than you originally made, when the mathematical implications of your original argument are the context under question, is literally moving the goalposts.
If you had said it was 5 flips in a row instead of 500 flips, then I wouldn’t have said what I said, because there’s a difference so wide between the two statements in their degree of probability as to be different arguments entirely. You didn’t say that though, and so I don’t know what you’re trying to say now.