Wouldn't that make it easier for someone that has access to hashed passwords in the case of a database leak? They would just have to submit the username and the hashed password (which they now have).

You're right, but the attacker won't get the user's original password that they probably reuse elsewhere.

If it's just your authentication system hashes that are compromised, the damage can be contained.

In this case client side will have our algorithm (i.e in JavaScript) + private key which we will use to hash the password. If this is the case I could not see any different between giving hacker password or hash-password with algorithm and key.

While there is merit to clientside hashing, you should always hash serverside as well, lest a leak prove catastrophic.