Perhaps not and you don’t wear running shoes with your dress suit as well, unless you are a Justin Timberlake fan.
Usually we have at least 2-3 types of shoes (could be more and one may have more than one of each type). Each for what it is meant for and based on what type of foot we have. There is no shoe that works for all type of feet, one that we can use for running, for wearing with suits and for our casual wear. If at all there were one, it would compromise on all fronts and risk your health.
Well, this is not a fashion or health related blog. So you know where I am going from here….