With private/public key pairs, the same scenario would result in the attacker only obtaining your public key, which is useless without the private key.
In general however, I believe that there is no inherit advantage of certificates over passwords, except for the key-size obviously. Everything else is just convention/standards.
Please see the following page that explains better what I meant when I said that the password should be hashed: https://en.wikipedia.org/wiki/Hash_chain
Using such a mechanism (including salts / challenges) will prevent an attacker using the hash as the password.
They're really not, they are private/public key pairs.
Now, sure, if you have a public key, and a guess at a private key - you can check if said guessed private key can derive the public key - but unless you know something no one else does - there's no practical way to enumerate and guess the key space.
But a password may be arbitrarily short and eg a random password of 8 mixed-case letters, numbers and some symbol is "just" log2((26*2+10+10)^8) ~ 50 bits (rounded up). Now that's probably not feasible to brute force on-line (without anyone noticing) - but quite feasible off line (if you're patient - but you don't have to be immortal).
I guess I should have said "Public Key Authentication" since that would be more reflective of a desirable baseline SSHD configuration.