No real puzzles work that way, so that means you're missing the reason why it's not a mate but a checkmate, or why it leads to checkmate (if it's a queen sacrifice to open up a key weakness in the king's position, the puzzle will generally require you to make the next move as well).
Got any examples of puzzles that you think are wrong in this way? Next time you come across one, post it here and people can explain it to you.
Anyways, for mine there are 2 fundamental flaws in puzzles.
1. You know there's a winning move, somewhere. So you keep looking until you find it. In a real game, you don't know that, so it's way easier to miss. The hope is that by doing the puzzle you'll recognise that pattern if the situation ever does arise in an actual game, but you need to do a lot of puzzles, and it may be a bit of a chicken & egg thing: you need to know the pattern already to solve the puzzle (at least, that seems to be what Hikaru does when he blazes through puzzles at the rate that he can & does)
2. They make you find the best move and won't accept any other winning moves. Who cares if you force checkmate in 4 moves rather than 2? If I miss a checkmate but win their queen, is that really that much of a big deal?
So many of the puzzles are designed so that you can only "win" by throwing your Queen at the opponent's King, with no defending pieces. You get mate, but not checkmate, and the king (in the real world) would just take the Queen and be like "Meh". How is that teaching good strategy?