Painting Roads on Terrain

Hello, I'm currently working on a stratergy game where you build houses and stuff. When you build a house in game the player should be able to connect it to other buildings with roads. My Idea was to paint these roads directly on the terrain using TerrainData.SetAlphamp but i couldn't get it working properly. My main problem is the low resolution of the texture you are able to paint. Is there any way to fix this. Does anyone have another idea to make (smooth) roads or even has a script for that? The roads schould look like this

Thank you for your response :)

Edit: What I currently have is a list of vector3 points the road should go through.