Hmm, grew up playing games placing pieces on the intersections (not just go, Chinese chess as well), the “grid” is never a unit of count to my eyes, but “obstacles” and background that need to be filtered out.
And counting territory or estimation is not something have to learn early on playing Go, but local patterns that are strong or weak, or in good shapes. In that regard, the lines helps with the pattern finding to me, since it’s easier to line them up strait or in different shapes without zigzagging.
Most stones are not big enough to cover up the intersections completely, even if they are next to each other (they have to be, or it would be difficult to place them and pick them up), with plenty of room showing the line beneath. And my brain can just see the lines behind the stones (like seeing a line even if it is a dotted line)