So the game would check against a blockchain that the signature is valid every time you want to play? What prevents you from just keeping a snapshot of the blockchain at a point when you owned it?
verification signatures don't use the blockchain, this is just public private key cryptography the same as PGP. the blockchain, for this purpose, just helped proliferate a standard application of that in a few years, where PGP failed at that for 25 years and counting as it has stayed very niche. many Web3 services use this for determining a user, or even accepting terms and conditions, while logging them in.
the client and the server would be checking nearest nodes for the latest block height. so any mismatch would prevent the user from keeping a snapshot of an old blockchain state as the max height would be too different.
it is also easy to disconnect a user if the assets moved when reading the current blockheight. many web3 applications just keep a websocket open to several nodes, this is boilerplate code at this point. reading from a blockchain is free. many services already check for onchain tokens to enable or disable access, for several years now. you can try gaming them to trick for access and see if you find anything.
