If Alice sends a symmetrically encrypted message to Bob, and Bob leaks the key, now every message Bob sends to Alice can be eavesdropped by third parties.

If the key is asymmetric, your only hope is to snoop on Bob while he is still writing Alice's key pre-encryption. He could, of course, deliberately leak every plaintext he sends - but if he is not malicious, and only leaked Alice's key accidentally, asymmetric nullifies the problem. Leaking private key is very bad. Leaking public key - who cares?

Because nobody cares that your public key is leaked, you might as well leak it yourself. If you leak it well enough, by broadcasting all over the net from public servers, you can even assume that everyone already has your key (or can get it). Now Alice's other problem is solved - how to get a key to Bob in the first place.

If Alice sends a symmetrically encrypted message to Bob, and Bob leaks the key, now every message Bob sends to Alice can be eavesdropped by third parties.

If the key is asymmetric, your only hope is to snoop on Bob while he is still writing Alice's key pre-encryption. He could, of course, deliberately leak every plaintext he sends - but if he is not malicious, and only leaked Alice's key accidentally (or maybe it was stolen and he doesn't know), asymmetric nullifies the problem. Leaking private key is very bad. Leaking public key - who cares?

Because nobody cares that your public key is leaked, you might as well leak it yourself. If you leak it well enough, by broadcasting all over the net from public servers, you can even assume that everyone already has your key (or can get it). Now Alice's other problem is solved - how to get a key to Bob in the first place.