If it is tossed right (no spinning like a pizza), it can be fair or it approaches fairness.

A person can come ahead in a coin toss game if they practice enough by adjusting the height, speed, angle of release, starting positions and etc to get the odds in their favor.

When I was a kid, my friends and I decided on decisions playing odds and evens which is much better than a coin toss. You can't practice or cheat with odds and evens unless you can read your opponent's mind of what they will throw in advance.

Everyone who has settled some trivial decision with a coin toss is going to be totally, like, incensed when they read this. Right?

Indeed. I suppose this, like many other similar discussions, comes down to whether the quality of "randomness" is good enough for the situation at hand.
For the sports games mentioned earlier it could be said that the quality of the randomness is important enough that some safeguards do need to be in place to retain fairness. I paid enough attention to coin tosses for games as I don't follow sports but I suspect that some basic precautions are taken to guard against the referee colluding with one of the teams (for example the referee not knowing the starting state of the coin).

If you suspect that the referee is colluding with one of the teams then the coin toss is probably the smallest part of the whole game where the influence of the referee matters.

What about holding the coin vertically and letting it fall to the ground? That would be close to fairness, wouldn't it? (Strange way to toss a coin, though).

But the thing which defines it as, "by definition, not 'fair'," is that it must be tossed "right." The outcome, therefore, depends not only upon the coin but upon the way in which it is tossed. Since that source of influence exists, the toss is not and cannot be mathematically-speaking "fair."

However, the "most-influential influencers" of the outcome of the toss are, indeed, "mostly random." This is why coin-tosses are used in sporting events.

That wouldn't work, because the coin is not evenly weighted. One side is heavier than the other and that side is much more likely to go face down. Tossing it in the air and catching it eliminates this bias.