Chicago faucets are better quality than T&S IMO,they are very hvy dutyIf it was me I'd tell them to keep it and I'd just rebuild it. Since you know it's a chicago faucet brand you could email them the pic and find out what seats, or at least the thread pitch you'd need.
But since you don't have much experience just replace it. The T&S will be slightly more durable and easier to get parts for. I assume the new CF you're looking at will be the quaturn stems which are really good too but don't take standard washers. The T&S can be upgraded/bought with ceramic stems. If there are no valves for just this faucet I'd suggest a model with integrated stops so you don't have to shut the main next time. The complete replacement T&S stems come with new handles, something that often fails on all faucets because people crank on them.
I'd go T&S.