I like this idea. As to correcting domination, I have some thoughts.

Not complicated at all, might work: Give everyone a random time, and just let it ride. Cross the ‘oh ****s, I broke it’ bridge when we get to it (and I’m not sure we will)

Sort of complicated, will probably work: Give people the ability to change their rollover time. Doing so will result in you getting no more turns until that time on the following day (so, at least 24 hours from now), and will remove you from domination for the current cycle.

Really, really complicated, will fix the problem forever: Domination counts all the visits from, say, the last 7 rollover periods. So, a domination cycle beginning Monday at 12AM EST (for example) would include the next 7 full cycles, including a 24 hour period to complete it. Since someone could, in theory, get their first turn in that round at 11 PM Monday, the round beginning monday would last until 11PM the following tuesday, but would not include turns granted on that tuesday. Told you it was complicated.