No. It’s not that the less-lethal form is more contagious. It’s that a live host with mild to no symptoms is able to pass the infection to more people than a severely ill host who’s first bedridden and then dead. And someone who has been infected with the less-deadly strain is likely to have immunity to the deadlier strain. More people get infected with the less-deadly strain not because it transmits more easily, but because it doesn’t sabotage itself by immobilizing or killing the host.
One complicating factor with SARS-CoV-2 is that it’s asymptomatic in so many people. It seems to me that would reduce the evolutionary pressure to evolve to a less-lethal form. That is, it’s already less-lethal for a lot of people.