Universal doesn't need to overcome Disney's reputation or guest appeal- they just have to offer a viable alternative.
The current model, as you say, is for Universal to offer an alternative day or two to families booking a mostly-Disney vacation.
With the addition of the water park, that changes fundamentally.
Universal will now have two 1.5 day parks, a waterpark, and a nighttime dining/entertainment district to justify families making a 4-day stay at one of Universal's four, soon to be five resorts, and never having to leave Universal property. That's the real game-changer, because Universal is about to go from having a mutually-beneficial relationship* with Disney to truly competing with them for full family resort stays.
*when I say "mutually-beneficial" what I mean is that I believe that Disney benefited from the opening of Wizarding World almost as much as Universal did; Disney opened almost nothing in terms of new attractions during the great recession while Universal closed out the recession by opening the most impressive themed lands America had ever seen. As a result, both Universal and Disney attendance grew during the same time period, again, likely because families were using the prospect of a day seeing Universal's new IOA offerings as a reason to book a return trip see the old stuff at Disney.