Yes, you are right, that is why i said its 'technically possible' because:

You don't want to be streaming the last 10,000 blocks to find 9 noughts-and-crosses moves.

It pretty impractical to do this on some public Node (Hive API). so in conclusion the game is decentralized and if you have doubts you can always verify running your own node.. a better way? maybe this tic-tac-toe game can have a ledger & blocks storage & hash verification (like hive-engine) which would be faster since now your only looking the game transactions (but a mumbo jumbo of blockchain code witch is totally op for a tic-tac-toe game), don't know if it makes sense. the final goal is that it can't be cheated, you can always verify.