It seems like the most obvious way to get people to care about professional level Overwatch is to get people involved. Currently, your casual gold or plat pleb, has no clue what's going on in organized overwatch. They've got an idea, but they don't truly understand what's going on. So, if you look at other successful esports and sports in general, the biggest fans are the ones who actually play. Take csgo for example, the guys I know on actual teams in ESEA main/open are the ones who watch every event and talk about them and follow every player and team.
So why doesn't Blizzard find a way to implement team building, scrimming, and eventual seasons like the ESEA divisions for all levels. Make it easy to form a 6 or more man roster by bringing players together outside of matchmaking and then make it easy for these teams to scrim each other. It'd be easy as hell. People put up an "ad" with their best heros or what position they'd like to play and captains or whatever can just look at player profiles and SR and such and start picking and trying people out.
Obviously, the OWL seems to have this in mind with contenders and such but it doesn't seem like they are making it easy for people to play truly organized overwatch. Especially not the majority player base.
Basically, in my opinion, currently it is extremely difficult to get involved in any sort of organized overwatch even at a super casual level. You've got seek out like minded parties in third party apps like discord or on websites.
Make it easy.