Why have the overhead of a second unique index? If it's "ease of joins", then I agree with Francisco Olarte and use the business logic keys in your joins even though it's a bit of extra work.
The strongest case, for me, when a surrogate key is highly desirable is when there is no truly natural key and the best key for the model is potentially alterable. Specific, the "name" of something. If I add myself to a database and make name unique, so David Johnston, then someone else comes along with the same name and now I want to add the new person as, say David A. Johnston AND rename my existing record to David G. Johnston. I keep the needed uniqueness and don't need to cobble together other data elements. Or, if I were to use email address as the key the same physical entity can now change their address without me having to cascade update all FK instances too. Avoiding the FK cascade when enforcing a non-ideal PK is a major good reason to assign a surrogate.
There's always the "account number", which is usually synthetic. Credit Card numbers are also synthetic. ICD numbers are (relatively) synthetic, too.
But that doesn't mean we have to use them willy-nilly everywhere.