If I get heads, I know the only possible way for the winning flip to be heads would be if both the other players got tails, whereas the winning flip will be tails if the other two got heads *or* if one got heads and the other got tails.
I agree with this, but I want to add a subtle point: it's correct to switch *even if I haven't looked at my own coin*. That's because, despite the fact that I don't know whether my own coin is heads or tails, I know that, whichever it is, it's more likely to be in the majority than the minority.
That's not to say that *nobody* needs to look at my coin. In order to determine whether or not my choice to "switch" puts me in the heads or the tails group, *someone's* going to have to look at my coin. But the rules of the game allow me to pass off this act of looking to someone else - they essentially allow me to tell the casino worker "hey, take a look at my coin, will ya, and assign me to the opposite". The important point is that I can safely pass off this instruction to switch without even knowing the result of my coin-flip, because I know that, whatever my coin is, it's more likely to be in the majority group.
If we change the rules of the game slightly, and say that, instead of choosing whether or not to "switch", you have to actually choose "heads" or "tails", then, of course, you yourself do need to see the result of your own coin-flip.