Well most of the time that interest is gone after the Preakness anyway. That can't be a good reason why you don't do it.
And if a horse wins the first two, the interest will still be there for the Belmont. The race where people could forget or lose some interest would be the Preakness.
You can run press over a three-week break. But you can't really do that for a single event over three months.
People complain all the time about the NFL offseason coverage, or ad nauseum discussion of particular athletes, and those are subjects they care about. Three months of a sport that so few people find regularly interesting nowadays, and people will go away.
Plus, even if there isn't a Triple-Crown attempt, the Preakness still goes almost right into the Belmont, allowing it to remain in the public consciousness. If you have a three-month break, only the 587 people who care about horse racing will come back.