I would guess the "stunt" here is that everyone knows that drinking one glass of Flint water won't really hurt you. It's the long-time exposure from childhood that is the problem.
If instead he starts in with a lecture about how it's cumulative exposure, affects children more severely, and that is why he won't take a drink as a publicity stunt, he knows that all that will be reported is "Obama refuses to drink Flint water!"
This wasn't a planned stunt in my estimation (unless you think the reporter was "in on it").
There are similarities with people who virtue-signal by working with children in a ghetto of a 3rd world country, and get photographed doing it - while actually living in the comfort of a luxury hotel and just showing up in the ghetto for a brief time. The intentions may be very good but people may still perceive it as hypocrisy. (Not particularly blaming Obama for this, he maybe in fact came to a situation where there is no good way out.)
But ignoring all of that -- why is it a good thing that he said the water was safe? The water wasn't safe to drink, and the video of the President sipping water from Flint means that the entire country collectively agreed that the situation was fixed. But it wasn't fixed -- the water in Flint is still contaminated with lead today.
[1]: https://www.youtube.com/watch?v=u2ZynkD3N_k [2]: https://www.youtube.com/watch?v=AjugN-nUHh8
The thing I'm referring to as being clearly staged is when he asks in the middle of a speech (where he was talking about things that need to be done in Flint) for some water, takes a tiny sip (he must've been really thirsty!) and continues. If you don't see that as being staged, I really don't know what I could possibly say to convince you otherwise. People from the crowd were shouting "don't drink" and he counters with "I know I'm going to be okay, because I've already had some Flint water."
And you're right that he mentions it's filtered (after being asked that in a follow-up and not when he first did the stunt, but whatever) -- but just because he's not outright lying about the situation doesn't make it the earlier stunt any less staged. There has been very little further outcry about Flint in the past 6 years, because most people you talk to will say "Oh, wasn't that thing in Flint solved years ago? Obama drank the water when he was there!
Was that not the case?
Looks like they're still available for free pickup: https://www.michigan.gov/flintwater/0,6092,7-345-73954-36627...
https://www.businessinsider.com/michael-skolnik-lead-poisoni...
Probably to laugh and say instead of "trying the water" he will see the water is sent to a lab for proper scientific testing and the results published to the public.
The general public doesn't really know that. People think that lead is this magical chemical that manically makes people stupid if they so much as lay eyes upon it. I guess that's better than thinking it's harmless but it's certainly non-optimal from a societal decision making standpoint. I would be very surprised if the reporter knew that the lead content of the water was mostly irrelevant to an middle aged man drinking one glass.