I'd have fully agreed with you a couple of year ago, but I think the role is becoming more and more specialized as someone who can do both defend and attack, excellently. Especially in formations for managers like Pep/LvG/Wenger etc. Mourinho would probably disagree with me a bit though
But if I think of the modern day fullback, or the best fullbacks out there, most are almost rated (more) for their attacking abilities, versus their defensive abilities. I'm not saying they aren't good defenders, but yeah. Someone like Marcelo for instance. He's constantly mentioned in the "best LB" discussions, over someone like Luis.
Alaba, Alves, Marcelo, Alaba, Lahm (before his move infield), Bernat, Coentrao, Zabaleta, Carvajal, Evra, Lichtsteiner etc., all very capable at both ends of the field (though debatably Marcelo's defensive abilities are average IMO).