It is important to improve voter engagement, which will translate, I believe, into improved voter turnout. But whether this is more- or less-important than improving the voting system is a false choice.
While a new voting system is important and can in the end improve voter engagement and hence turnout, it will not, suddenly, magically, turn things around over night. It will help, in time, as people see that their votes are more meaningful, and have a higher sense that the winners are legitimate, but there is no silver bullet.
Be that as it may, however, the choice here is NOT between achieving a new voting system, vs improving voter engagement. These are to some degree different problems that can be, and should be, addressed separately, though they can still be addressed concurrently.
We can still encourage debate and town-halls, explore social media for getting people involved, recruit interesting candidates, knock on doors, and consider things like online voting, and so-on – quite independently, as well as at the same time improving the fundamental voting system.