I've spent my entire golfing life (36 Years) trying to get comfortable with flat surface putter grip. I've tried every size and shape you can get and I've always struggled with pushing putts and with leaving them short.
I saw a test on MGS where different putter grips were tested and noticed that the best face angle and path numbers were using a round grip. The make percentage was the lowest, but that seemed like an alignment issue to me. I put a Pure DTX Midsize on my putter, put the laser on the floor and starting putting balls straight down the laser line. I started testing other round grips and sizes and ended up with a CP2 Pro Jumbo on my putter.
I still grip it like a putter, through the lifeline of the left hand, and let the right hand get comfortable. Find the line, square the face to the line and hit the center of the putter face.
Never putted better!!!