I used to have a difficult time putting, specifically, getting the ball to start on the line intended.  Although, the club face has the biggest effect on starting line, the club’s path can have a huge effect on the club face!
There are too many people who have their arms far away from their body where they have no support from the core, just flailing around.  The core will provide support and stability and help create an excellent path.  Think of this scenario as similar to carrying a serving tray out of the kitchen… clearly your elbows will be against your side creating the support needed.

Now let’s use this drill to help feel the connection to the core….  Stand vertically and rest your lower body against a chair or half wall.  Please hold the putter straight out so that the shaft is parallel to the ground and then slide the putter towards you until your elbows rest against your sides.  They should not be pressing into your side but kept soft.  Now rotate your shoulders around your spine while keeping the shaft parallel to the ground.  There will no longer be any tilting of the shoulders, only pure rotation.  You should feel your elbows sliding against your core.

Next you should move into good posture with minor flex of the knees and a good hip hinge while keeping your shoulders back so you don’t get too rounded.  Your arms/club should fall down to the ground while your elbows maintain soft contact with your sides.  Now use the elbows and your core to guide your stroke to a perfect path!