Wingers aren't 'dinosaur' football if used properly, that's a real misconception because Barcelona shy away from out-and-out wingers and they are the fashionable team.
Real Madrid, Bayern Munich, and Borussia Dortmund use width and are superior football teams to us. They are a good option and can co-exist with a technical style of football (like Real can have Di Maria and Ronaldo - width, in the same team as Ozil and Alonso - technical). Also I think wingers are extra useful in England, English defences seem to struggle against wingers which is why United have used them for over 20 years without being sussed out.
Even Barcelona have used Pedro, Sanchez, Tello, and others who are fast and play out wide, even if not always as traditional wingers. We have Scott Sinclair and James Milner...
If you choose not to use wingers you still need pace in other areas, our midfield has none. There's absolutely no spark in our midfield, we have no central midfielder like Vidal or Fernandinho who is all over the place and no attacking midfielder who can scare opponents like a Hazard or Bale. Only when Yaya goes into box-to-box powerhouse mode do we see it but that hardly ever seems to happen anymore. This is why our football just seems to... stop so often. That undeniable stagnant feel.
Silva is the only high-end creative outlet in our entire squad thanks to Nasri's poor performances. Tevez does a decent job as a pretend playmaker but that isn't his ideal role.
When people say add pace to the team, they don't always mean speed merchants on the wing. And it doesn't just mean running fast, but speed of thought and passing... Xabi Alonso doesn't have much pace but thinks so quickly and passes so well from deep that he can create a chance from the halfway line in a second. The best teams can create REAL chances (not these scrappy half-chances we so often make in tight spaces) from all over the pitch, as far back as central midfield.