Not a Londoner but try to visit whenever I can, and the Heathrow Express train to Paddington is the only way I will travel from the airport into Central London. The train ride itself is 15 minutes, which is faster than the tube, taxi or bus (I have taken them all in the past). Once you get to Paddington Station, you will have to make 2 transfers on the Tube to get to Leicester Square. So you can always taxi or Uber from Paddington if you don't want to do that with luggage (I don't recommend).
Otherwise, yes, Piccadilly Line does go direct to Leicester Square, so that's the easiest. But it's rather slow.