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Checkmate still exists in games that finish by King capture, as the condition where it is impossible to avoid such capture on the next move. It just doesn't terminate the game. Since at any serious level of play people would not blunder away their King, the only way to capture it would be by first achieving checkmate (or stalemate). So one could say that the goal is to achieve checkmate, so that you can then forcibly capture the King. 'Goal' doesn't necessarily mean terminating condition of the game; you can have many different strategic goals. So I don't see a very large contradiction here.

That it would be mandatory to declare check doesn't mean that it is mandatory to resolve it. Of course not resolving it means your King will get captured, and that sort of makes it mandatory enough for most players, even when the rules do not strictly demand it. The fact that check has to be declared makes it even less likely they would allow the capture by oversight. (Of course they could still move themselves into check by mistake; they would receive no warning against that.)

The special status of check is not really fundamental; it is a consequence of the peculiar way that Chess handles illegal moves. In Shogi any illegal move counts as a loss, and then it makes no difference at all whether moving into check is forbidden, or just a bad move that gets your King captured. In Chess it would decide whether you have to take back the move that exposed your King, and continue the game from there, or that the opponent can capture it for the win.