I went from the beginner putter ( not starter but beginner) straight to the level 71 Nike.
I thought i was ok with the beginner, but the difference is remarkable, so yes a better putter will help, but that is only the beginning. You need to learn how and where u must land the ball for the easiest putt.
That is quite simply the quickest way to improve, but it takes effort and patience to learn every course and every hole.
For example above or to the left and right of Kiawah holes is not good, by default stay below the hole.
That is generally a good rule for all links courses for some reason, so RSG anD STA the same