No, I think blockchains provide many useful functions. However, providing a token with value is a necessary feature. I’m very excited about things like Rootstock which add smart contract functionality to the bitcoin blockchain.
If you can find some other reason for processing transactions and can guarantee that the population is big enough then the coin becomes unnecessary.
An extrinsic reward doesn't have to be financial though, it's anything that you want to get out of an activity that isn't part of the activity itself.
If the blockchain ledger was a part of a greater whole and the client software of which it forms a part was somehow intrinsically fun, useful or needed, the token itself wouldn't be necessary.