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Except that the payout for those bets are generally done so that the house takes a cut of the overall action. The vig is baked into the payout for sport outcomes; betting on all the outcomes equally isn't going to probabilistically give you the payout equal to what you'd buy in.
Exactly. That's why I'm differentiating between games of skill (i.e. sports) and purely statistical "casino games". It's possible to beat the house in sports betting, but only if you are genuinely better than the house at considering the odds. Of course, the house will always try to set the odds in their own favour, but it's impossible for them to know the exact odds. Thus, a well informed player can, in principle, identify the games where the house has under-valued an outcome and exploit those.
This basically boils down to the fact that in a casino game, the probability of every possible outcome is known exactly, so the house can trivially set a payout that benefits them. In sports, it's impossible to know the exact probability of a given outcome, so the house can make mistakes.