What would happen had we not invented asymmetric encryption?

Without asymmetric encryption, the Internet as we know it would not exist, period.

• You can’t sell stuff to random client systems with symmetric encryption. Why? For symmetric encryption to work means that you and your intended recipient both must know the key and keep it secret. So as a merchant ideally you have to have unique secret key for ALL your customers, plus transport and store it somewhere securely. Otherwise, if someone else knows the key (say another customer), then they can decrypt your data and read it, which makes the entire encryption process pointless.
• With asymmetric encryption as a customer you just have to remember only public keys, and that is easy because public keys are public, so they can be “remembered” by being published somewhere, where everybody can see and use them. The Internet without asymmetric encryption would be like TV, where you look at the infomercial but you have to call in on the phone to make the purchase and verify you call to the right place.