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.
I've just always thought it a bit of a classless act from Tywin. They sent the man's bones back, but not the house ancestral sword.
Your comment is likely right, it's just never sat well with me and I'd be interested to know how it went down across the rest of the realm. Even in war I figured the great houses would have kept a modicum of respect for one another, or maybe I'm just naive.
In addition to the other commenter, I think you're exactly right, it was in poor taste, but it is also a storytelling device to show that the Lannisters will do anything to further their climb to power and their desire for legitimacy. I think it also has some world building value as it shows that, 1) a mark of a true great old family house is to have a sword of valaryian steel, 2) the starks are a great house, 3) the Lannisters are not, because they have no valaryian sword, but 4) they are willing to do underhanded things to obtain such an honor. It presages the downfall of Stark and the rise of Lannister, and a changing of the times, because it was a very ignoble act, but no one spoke up or tried to stop them.
The Lannisters used to have a Valyrian steel sword of their own named Brightroar. That sword was lost though.
It had always irked his father that none belonged to House Lannister. The old Kings of the Rock had owned such a weapon, but the greatsword Brightroar had been lost when the second King Tommen carried it back to Valyria on his fool's quest. He had never returned; nor had Uncle Gery, the youngest and most reckless of his father's brothers, who had gone seeking after the lost sword some eight years past.- ASOS - Tyrion IV
Tywin had been trying to get a replacement for many years with no luck.
Thrice at least Lord Tywin had offered to buy Valyrian longswords from impoverished lesser houses, but his advances had always been firmly rebuffed. The little lordlings would gladly part with their daughters should a Lannister come asking, but they cherished their old family swords. - ASOS - Tyrion IV
Acquiring Ice must have thrilled Tywin. Not just a sword to replace Brightroar, it was a two handed sword. It was large enough that, when melted down, it could supply enough Valyrian steel to make two swords. Now the Lannisters who had been without such a sword for so long would own *two *of them. Politically, Tywin likely considered the possesion of two Valyrian blades to be boon for his house. What other house has two such blades?
Morally... well, Tywin doesn't really care about morals. He cares about appearance, and the appearance of power above all else. This is a man who publicly despised Tyrion's whoring, but actually had a tunnel dug to a brothel so he could visit without being seen. This is also a man who killed every man woman, and child of house Reynes because they wouldn't pay their debts to house Lannister and had the gall to rebel.