Winchester supreme bullets you could use are the Failsafe and CT Accubond. The Failsafe is an excellent bullet and will work in either 140 or 160 gr versions for your needs. This bullet is moly coated, so I suspect this is where the "mess up the barrel" comment originates, as it will coat your bore in moly (this is said to reduce friction and improve function). If you plan to shoot moly bullets the first thing you need to do is ensure all the copper is removed from your bore prior to firing the moly bullets, this will provide better results. You must use a dedicated copper remover such as Barnes Cr 10, Sweets 7.62, etc.

The CT Accubond is lubalox coated, which is really nothing more than a paint. It has no adverse effects to your bore IME. I would only use a 160 CT accubond since you will be chasing elk!

Federal premium has more offering you can choose from of which none of the bullets are coated if that causes concern. Your choices are Nosler Partion, Nosler Accubond, Barnes TSX and TBBC. I'd probably stick with 160 version as suggested by your grandfather, but any of these bullets will work for your needs.

As always the best option is T&E than pick the best shooter in your rifle for your application.