Decoding is not verifying
Anyone can read a JWT - the payload is base64, not encryption. What proves it came from your server is the signature, and checking that needs the key. A tool that only decodes tells you what the token claims, not whether the claim is true.
Why alg "none" is flagged
A token whose header says alg: none carries no signature at all. Libraries that honour that header accept a token anyone could have written - the oldest JWT attack there is, and still found in the wild. This page marks it as an error rather than decoding it quietly.
Never paste a private key
RSA verification needs only the public half, which is published by design. Nothing legitimate asks you to paste a private key or an HMAC secret into a web page you do not control - including this one, which is why the whole check runs locally and the page says so.