We need a LW more than a RW because in Jose plan, the one responsible for providing the width on the right is Valencia not the RW. It's not a disadvantage of players, it's because Jose wants the RW to join the midfield and give Valencia space to rush forward and cross. Mikhi, Mata and Lingard can do this job but on the other side, he wants the LW to stick on the flank to provide width, especially when playing a LB that is only concerned with defense behind that LW. He tried Martial in this job and he failed as the only thing he does on the flank is to try to beat the next defender till it fail. Martial style of play is suited to cutting inside, which isn't what Jose wants him to do especially when giving the RW instructions to cut inside and joint the midfield. This devoid us of any width bar Valencia.
If you are going to play with Jose's classical 4-2-3-1, then yeah signing a LW to provide width makes perfect sense for me more than RW. The debate is whether Perisic is good or not or what we can find a better replacements for him for the job Jose wants him to do on the left, but in the tactical point of view, we do need a LW that always provides width to the team and allows the LB to set back defensively as Jose wants, otherwise, we'll get stuck with all our players crowded in the midfield with r trying to cut inside with only Valenica as a flank player, and this gives us troubles against any parked buses.
Searching for such a player makes perfect sense, whatever Perisic is up to it or not, Jose thought he is up to it then let's see.