Automatically spawn a ControlMaster background process on first access to a ssh remote system

You should use the ControlPersist config option.

 ControlPersist
         When used in conjunction with ControlMaster, specifies that the
         master connection should remain open in the background (waiting
         for future client connections) after the initial client connec‐
         tion has been closed.  If set to “no”, then the master connection
         will not be placed into the background, and will close as soon as
         the initial client connection is closed.  If set to “yes”, then
         the master connection will remain in the background indefinitely
         (until killed or closed via a mechanism such as the ssh(1) “-O
         exit” option).  If set to a time in seconds, or a time in any of
         the formats documented in sshd_config(5), then the backgrounded
         master connection will automatically terminate after it has
         remained idle (with no client connections) for the specified
         time.

ControlPersist no is the default behaviour, which is as you describe. I use ControlPersist 4h to allow the background sessions to clean themselves up periodically.