This has nothing to do with lightening network.
Exchanges store all bitcoins in their own wallet and only display a number as balance for each of their users. Exchanges can allow user A to send user B some fund without making a real transaction.
Let's say users A and B each have 1 BTC on an exchange and user A wants to send user B 1 BTC. Since the fund of both users are stored in the exchange wallet, there is no need to send bitcoin on the blockchain. So, the exchange decreases balance of user A to zero and increases balance of user B to 2 BTC.
Keep in mind that, those who keep their bitcoins on an exchange are not the real owner of those bitcoins.
Ok this make sense why one should not store btc in an exchange. And it is best to have a bitcoin wallet itself which generate bitcoin public key and its private key and can add public key to that certai wallet. The seed phrase of wallet is one of the important feature of thar wallet that should not be lost. Anyway, OP should not force everyone to join cryptocurrency. The blame will going to put in him especially in this high risk investment that will have the chances lose in it. If I were OP I can't handle someone blame me because I was the one introduces the bitcoin which let them failed.