Is it because I decided to send it to an exchange wallet in order to avoid the network congestion then?
Exactly because exchanges don't add BTC in your balance as long as they aren't confirmed.
3. Treat Whirlwind as a network and respect the rules, in this case we will not be able to help in any way and the user would lose access to his funds assuming it's true that indeed he lost the private key
I would probably have done this. You have no way to know whether he is speaking the truth and you can't identify whether he will abuse the note (he may be a liar). If you allow this, it's very possible that he is trying to scam you. You will never know whether he is a liar or not.
If the amount is too small, I would say give it back to him. If a big one, don't, and make sure you warn the people on the website that there's no chance of getting back any funds if they don't have certain information. As said above, it will create room for scamming you.