Is my node still useful even if it has never relayed first a transaction on the blockchain?
This sounds like something you have seen on blockchain.info. Their information hardly matters, they are just another node (with plenty of info about their state and a wallet service).
Your node constantly verifies and relays transactions for others. Open the debug.log (Help -> Debug Window -> "Open") and search for
ERROR: AcceptToMemoryPool: to see which you did not relay. Enter
getmempoolinfo in the console (Help -> Debug Window -> Console) to see how many you have relayed and are valid according to your nodes settings.
Are pruned nodes good for the bitcoin network even when they never relay transactions?
Yes, a pruned node still verifies and relays new blocks and all transactions. You cant help new nodes sync though and you cant currently use a pruned node as a wallet, but it still supports the network to run one.