Ssh – Seahorse / GNOME Keyring does not recognize the new ed25519 SSH key, is there another way to have the key unlocked upon login

The SSH agent handles signing of authentication data for you. When authenticating to a server, you are required to sign some data using your private key, to prove that you are, well, you.

As a security measure, most people sensibly protect their private keys with a passphrase, so any authentication attempt would require you to enter this passphrase. This can be undesirable, so the ssh-agent caches the key for you and you only need to enter the password once, when the agent wants to decrypt it (and often not even that, as the ssh-agent can be integrated with pam, which many distros do).

The SSH agent never hands these keys to client programs, but merely presents a socket over which clients can send it data and over which it responds with signed data. A side benefit of this is that you can use your private key even with programs you don't fully trust.

Another benefit of the SSH agent is that it can be forwarded over SSH. So when you ssh to host A, while forwarding your agent, you can then ssh from A to another host B without needing your key present (not even in encrypted form) on host A.

You solved part of your issue with the use of keychain to act as storage for your passphrases for your SSH keys. As to your other question:

However, I am not sure if it is worth to encrypt a private key which is only used for ssh connection to a specific server. Is there any concrete security risk by not encrypting the private keys? In principle if one is able to access the private key he/she is also able to use the passwords stored in the keychain...so it seems that there might not be a big advantage in encrypting the private keys

It's not that your private keys are left unencrypted, it's that they're not protected with a passphrase. Therefore anyone that can gain access to your home directory's .ssh directory can get control of your SSH keys without issue.

If you're comfortable with leaving these keys on a system that you have full control over then I see no issue with leaving them as passphrase-less. I would also recommend generating a different set of keys that you can leave on secondary systems such as this, or even a unique set just for this server, so that you can completely abandon it without much issue if you ever discover its been compromised.

As to using keychain to manage your keys, I see not much to be gained with having passphrases on SSH keys if you're using keychain to manage them, except that keychain will only store them when you use them. If you don't use the keys often, and are diligent about cleaning up keychain when you're finished, then a passphrase would be an advantage, but IMO, it's a small one.