Erase to intersection

Automatically, no. You have to do it manually. If it's a pencil, pen, or paint brush stroke, you can just click where the part you want to delete connects to the intersection and it will select a single line segment that you can delete. Once you break the connection to the intersection, you can double-click the remainder and it should select everything... assuming they're all connected.

But if you drew with the brush tool, the "line" is actually a vector outline, so no intersection even exists.