The main point is that slavery is officially illegal everywhere.
I think it's a spontaneous necessity for local population to thrive when they gain education and skills. It's more profitable for the society as a whole to avoid enslave doctors, engineers, lawyers, etc. (Now that I think about it, it sounds like a pretty good technique against brain drain)
What happens in practice in each country is another problem. There can be all sort of incidents of people trying to illegally take advantage of others that may or may not be close to slavery.
Like bonded labor that comes with the justification of debt to get the moral highground. It allows you to decide arbitrarily how much your debtor earn and leave them little enough so they're forced to borrow more from you lol.
In the case of illegal slavery, there's little chance that slaves are well treated and used for high educated skill like it could happen in ancient rome.
Honestly though, it's not the west concern. On the contrary, western companies would gain more to enslave labor and be as untransparent as possible.
Maybe western individuals will say they care about slavery in third world country, but the truth is that they don't care enough to face armed angry gangsters defending their business lol.
Maybe Prince said the west are the only one who want to end slavery because he was thinking the universal rights and the UN were originally a western thing?