Stipend doesn’t always increase after passing quals. It depends on the program.
At most programs your funding is contingent upon your academic progress and the availability of funding. Even if you chose UConn, where the funding is ‘guaranteed’ for 5 years, if you don’t maintain a B or higher in every class they can kick you out or yank your funding. And obviously if the program loses money or significant grant funds and needs to make cuts to current students’ funding, they can. These things are somewhat unlikely to happen especially at a top program, but Stony Brook is just being extra cautious about warning you up front.
What you should do is call the graduate secretary and ask her/him how often students are fully funded by the department for the X years it takes them to finish. No, this is not rude - this is smart. Most likely she’ll tell you that students are funded for years 2-5+ as a matter of course unless they really go off track.
And yes, program reputation matters a lot. There’s a big difference between a top 25 school and a top 100 school in academic hiring.