It was a symbol of power and added legitimacy to Starks by tradition. Remove the symbol, and you weaken whatever it stands for.

Also, Valyrian steel was pretty much impossible to make anymore since the secrets of refining the metal was lost with valyria, which made the sword pretty much priceless. Some master blacksmiths knew how to work it, but not how to make the steel. So it was logical to add a few of these priceless blades to the Lannister arsenal, and melting the original down would help remove the link to the original owners and legitimise the new ownership.

I suppose they could have not reforged it and just renamed it or something, but these weapons as rare and revered as they are, will have records of it being used and by whom - a paper trail.

Anyway thats what I think.