TL;DR: carry a true random number generator with known distribution on your person at all times[*].
If there's no correlation between your assignment of outcomes to faces of the SD card, and the SD card's bias, then yes, it doesn't matter whether the SD card is biassed. Saying "no correlation" means it's as if the outcomes were assigned to the faces of the SD card by an unbiassed random choice. So there's your randomness, regardless of what the SD card does. It could be 100% biassed as long as you assign the outcomes to faces randomly. This isn't disturbing :-)
Naively one can argue that if we're capable of assigning outcomes to faces "as if randomly" then we could just choose an outcome "as if randomly". Not so, of course, the point is that any knowledge we have of which outcome will actually happen affects our assignment. For an unbiassed coin flip we cannot have any such knowledge. For a biassed coin flip we're concerned that we might unknowingly act on any information or prejudice we do have. Note that even if we're in some sense trying to get a particular outcome and our information/prejudice is wrong, we still create an unfair choice. It's unfair in the opposite direction to the one we want, but still unfair. That's enough to lose at rock-paper-scissors, and surely the ultimate goal of all probability and game theory is to break even at rock-paper-scissors?
So, you could argue against the existence of that correlation in two ways:
- Experiment. Observe whether you in particular, or people in general, seem to create any bias of outcomes.
- Theory. State that since you have no knowledge of which side of the card is favoured, you cannot possibly correlate that to the choice of outcomes.
Experiment 1 seems overkill for this particular case, since the actual outcomes matter -- there might be some outcomes that your brain is capable of assigning in an unbiassed way and others which it is not. A more general such study might be good. Besides, if you could set that up you could probably just fetch a coin.
Theory 2 disappears as soon as you have used the SD card once. At that point you have some information (admittedly not a lot to start with) about its possible bias, so you can no longer claim to be ignorant.
In fact I think argument 2 is dubious anyway. Aside from anything else it's possible that you are subconsciously able to observe from the SD card some feature that (without you realising it) contributes to its bias when flipped. Furthermore that could affect your assignment of outcomes.
If you couldn't see the SD card then you'd be on firmer ground. Suppose that a third party coloured the faces of the SD card "red" and "blue", and you assigned the outcomes to "red" and "blue" before seeing the card. Then my proposed connection between flip bias and outcome bias needs two correlations:
- you need to assign the outcomes to "red" and "blue" based on some intuition of which is more likely
- the third party needs to assign "red" and "blue" to the faces based on some intuition of which face is more likely.
If either of those fails, then the choice of outcome becomes fair.
It starts to sound implausible both those things would hold, but the fact I'm even considering it is because I consider the original scenario plausible, i.e. your original process is not certainly free from bias.
For the football game where the coin is later discovered to be biassed, then provided everybody accepts that there is no way that bias could have affected the call or the person making the toss, I don't think there's a problem. But probably in practice everyone would accept that on somewhat shaky grounds. Good enough for football isn't necessarily good enough for mathematical theory.
[*] von Neumann's construction of a known distribution from an unknown distribution is fine, provided that you can't somehow (intentionally or unintentionally) influence the outcome of the second toss based on the result of the first. You can achieve that in the first round by using a camera, and only examining the result of the first toss after making the second. However if you get a pair and go to a subsequent round then you risk having some information about how to effect that result, which could alter your flipping action. Basically when you don't trust your own subconscious influences you're in deep trouble.