Using the ergopad.io website is not the same as using the Ergo blockchain. When you use the ErgoPad website to generate transactrions for you, you'll still need to use a native Ergo wallet to send the transactions and interact with the blockchain.
Connecting your wallet on the ergopad.io website doesn't give the website any control over your funds. It stores the public address of your Ergo wallet in localStorage on your device, and will use that data to compile ergoscript contracts for you to interact with on chain. The data will never be stored on the ergopad.io servers.