I don't think there's been much of a media circus with Kawhi currently, and I don't think there was much of a media circus last year with Paul George. If the the C's get AD, I don't think it will be that bad. There will be a game or two (like when the C's play in LA) that may get some questions, but it won't be anything out of the ordinary.
The only reason there's been a media circus lately is because several new dominoes fell all at once right around the trade deadline (AD trade request, Porzingis trade, etc.). If the C's trade for AD, there's not really going to be any new dominoes to fall next season that will re-ignite rumors.
LAL for instance, we'll probably know their cap situation for the summer of '20 by the start of next season, so there's not going to be a surprise trade (like Porzingis) that suddenly clears cap room so they can make a run at him, adding new speculation. The only way that would change is if LeBron has some devastating injury or gets traded (or if AD changes agents again), and then that will bring up questions if AD still wants to go to the Lakers now that LeBron has torn his ACL and been traded to Memphis.
There'll be a few questions when he gets here, occasional questions during the season, and you'll get the same vanilla answers players always give: "I'm not even thinking about that, I'm just concentrating on this season."
No circus.