I think there can be a slightly different nuance between skilled and skil(l)ful. The former more often refers to a set of skills formally acquired for work. Britain used to class people into socio-economic groups such as skilled labourers, semi-skilled labourers, etc., and it would be more natural to say 'skilled workers' than 'skil(l)ful workers' in an economic or business context.
However, regarding the surgeon, both are correct English.