I was thinking more about a SR-specific tournament, whether local or national, and I just don't see how it could happen - and what problem it would be solving at the local level. Right now, at both the younger levels, and lower levels of competitiveness, teams organize themselves in to brackets. A club often tries to have a team in several brackets at each age group. It's often Gold/Silver/Bronze, or Flight 1/Flight 2/Flight 3, or any of the other combinations that we're all familiar with. A team is in one of those color-coded bracket for league, and tournaments as well set themselves up similarly. Tournament directors try and bracket teams together, so there is one for Gold teams, one for Silver teams, etc. It's a simple/straightforward setup that has been used for decades.
Now - a savvy tournament director could look at the brackets they've set up, see the SR ratings of the teams, and realize that in their particular case that weekend, the brackets are actually from 38-41, from 41-43, and 43-45. Naming the brackets that way would be silly, but it allows them to validate whether teams are actually in the right place. When they see that a team that's actually a 45 is sitting in a bracket with the rest of the teams from 38-41, they have a pretty good idea that the team will steamroll the division - regardless of what color they are calling themselves. In that case the tournament director has time to work with the coaches to figure out matchup that will results in closer games / less blowouts / better competition.
That's what's happening today whenever a tournament director uses SR to help their bracketing. It's what could be happening more tomorrow as they become aware of ways that it would make their lives easier. As I understand it, if a tournament director sends a link to the teams to SR, they will help develop the brackets to seed teams the closest - but I have no information on how widespread this is, and whether it's 5-10 tournaments across the US, or 100+.
None of the leagues/tournaments need to become more SR-branded, or really make any public-facing change if they choose to go down this path. The only result is that there should be less blowouts. SR used to rank tournaments by fairness of competition, and showed both those tht were very fairly flighted, as well as highlighting those tournaments that were very poorly flighted, resulting in way too many mismatches.