If Milwaukee wants another lottery pick this Draft (Miami's #13), then they have to do the Giannis trade before then. Otherwise, they risk losing out on an asset like that and also, I think the level of interest and quality of offers from other teams will dwindle too. Most teams including Boston and Miami probably have Free Agency plans and other plans in place as well, and I doubt they'd want the Giannis situation lingering and potentially missing out on other things because of the Giannis situation. And Giannis is a FA after next season, a team like Miami can just wait and sign him outright if they wish to then.
I don't think the Bucks have much leverage. Sure, they could wait it out, but at what risk? Giannis could just say he doesn't want to play for them next season and sit out for a while. Meanwhile Miami could say, "whatever" and just wait until next summer. MIL won't extend Giannis either if Giannis says he wants to go elsewhere.