How to properly git clone with public key from gitlab?

I have a gitlab projet on our local Gitlab instance configured and my account is set with a public key.

Strangely when I try to "git clone (...)" a project for which I have the correct permission from a client host, gitlab asks for a password even if it should pick my SSH key, no matter what is configured in my client SSH config file.

git command

git clone git@my_host.tld:username/project.git
Cloning into 'project'...
git@my_host.tld's password:

~/.ssh/config

Host my_host.tld
        Hostname my_host.tld
        IdentityFile /root/.ssh/gitlab_key
        Preferredauthentications publickey

When I try to ssh using a regular command, it works:

ssh command

ssh -vT -i .ssh/my_host.tld

debug1: Next authentication method: publickey
debug1: Offering RSA public key: .ssh/gitlab_key
debug1: Server accepts key: pkalg rsa-sha2-512 blen 535
debug1: Authentication succeeded (publickey).
Authenticated to my_host.tld ([XX.XX.XX.XX]:22).
debug1: channel 0: new [client-session]
debug1: Requesting no-more-sessions@openssh.com
debug1: Entering interactive session.
debug1: pledge: network
debug1: client_input_global_request: rtype hostkeys-00@openssh.com want_reply 0
debug1: Remote: Forced command.
debug1: Remote: Port forwarding disabled.
debug1: Remote: X11 forwarding disabled.
debug1: Remote: Agent forwarding disabled.
debug1: Remote: PTY allocation disabled.
debug1: Remote: Forced command.
debug1: Remote: Port forwarding disabled.
debug1: Remote: X11 forwarding disabled.
debug1: Remote: Agent forwarding disabled.
debug1: Remote: PTY allocation disabled.
debug1: Sending environment.
debug1: Sending env LANG = xx_X.UTF-8
Welcome to GitLab, @username!
debug1: client_input_channel_req: channel 0 rtype exit-status reply 0
debug1: client_input_channel_req: channel 0 rtype eow@openssh.com reply 0
debug1: channel 0: free: client-session, nchannels 1
Transferred: sent 3600, received 3464 bytes, in 0.5 seconds
Bytes per second: sent 6808.3, received 6551.1
debug1: Exit status 0

What have I missed? How to make git pick my SSH key instead of the password authentication method?

Expanding my comment to an answer:

You're specifying /root/.ssh/gitlab_key in the posted ~/.ssh/config. User ~/.ssh folders should only have mode 0700; this means that only that user can access the folder or any of its contents. I'm presuming here that you're running this command as yourself, and not as root, because you shouldn't use root for day-to-day work.

At minimum, you'll need to update your ~/.ssh/config to reference your own key - the one that your ssh -Tv command worked with:

Host my_host.tld
  IdentityFile ~/.ssh/my_host.tld