It seems that
ssh-add -K ~/.ssh/id_rsa will load your key but will ask for the password each time you reboot.
I am looking for a solution that would not require me to re-enter the key password between logins.
You are being prompted for the password every time because your "login" keychain is being locked after inactivity and/or sleeping or in your case a reboot. There are two ways to solve this for you.
You should not be prompted for a keychain password anymore.
Add the public key in:
Public key usually are on:
Hope that helps