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From the article: "We will be chiefly concerned with the middle game and will not consider the end game at all."

The effect of having the checkmate rule instead of capturing the king is allowing the possibility of stalemates. But stalemate is rare in the middle game, where there are usually many legal moves available, so it's simpler to pretend the goal is just to capture the king.

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It is however important to note that allowing kings to be taken can result in both kings being traded which by the scoring function would be considered neutral. This possibility does not occur in normal games so a search with that modification may generate the wrong value.

Games would need to also be stopped when a king is taken to make the search approximately correct.

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