I use the grid lines to line up the edge of the fabric on and double check the ruler measurements against the grid lines before I cut. I too know the lines of the rulers are the same as the grid lines. My sewing or cutting hasn't been off unless I was using the wrong measurement on the pattern. (Ugh! more than once!)
I line up the straight edge of the fabric. Then I use one ruler on the right side (I am right-handed) to line up my measurement needed along that straight edge of the fabric. (The left edge of the ruler will be onto the fabric.) I make sure the fabric edge is exactly on the measurement of the ruler. Then I take another ruler on the left side and butt it up against the ruler on the right. Double-check the measurements up and down the rulers against the grid lines on the mat for accuracy, slide the ruler on the right out of the way, and cut the fabric on the edge of the ruler that was on the left. Becomes habit to use the ruler markings and to use two rulers for accurate measuring and cutting. When I do it this way I can place the fabric anywhere that is convenient on the mat. Doesn't matter how big of fabric pieces I am cutting, I use 2 rulers for measuring and cutting. For me, the mat grid lines are just for straightening up the fabric, but all the measuring and cutting is done using rulers. I learned to do it this way because lining up the fabric to the 0 marking on the mat drove me nuts. May sound anal but I have to have the measurements on my right, because I am holding down the ruler with my left hand.