I was sad when the Dreamcast died, but then one day grocery shopping I saw OXM with Panzer Dragoon Orta on the cover, and I had to have an XBOX. I could give a crap about the PS2.
Xbox Nation Magazine (the unofficial Xbox mag) was the reason I got an Xbox post-Dreamcast. I was sad to see Dreamcast Magazine cancelled, and doubly sad to see the Dreamcast dead. But then I learned that a lot of the Dreamcast Magazine crew went on to Xbox Nation, and took with them their love for SEGA. As such, JSRF and Orta got awesome cover stories. At that point, Xbox just felt like such a natural next step. The console had an awesome magazine with a SEGA loving staff, *new* SEGA games (PS2 had only Dreamcast ports and arcade ports), a familiar controller layout, and emphasis on online gameplay.
Honestly, looking back, the PS2 was disappointing when it came to SEGA games. Yakuza, Yakuza 2, Virtua Fighter 4 and Virtua Fighter 4: Evolution were the standouts. Looking through the list of PS2 games I'm seeing Aero Elite: Combat Academy, Astro Boy, Shinobi (all poor titles), Red Entertainment published stuff, Dreamcast ports like Crazy Taxi, Rez, Headhunter, Space Channel 5, and a load of games that also appeared on Gamecube and Xbox (and in almost every instance the PS2 version was the worst).
Gamecube benefitted by having some exclusive titles, like Beach Spikers, Billy Hatcher, F-Zero, Phantasy Star Online Episode III: C.A.R.D. Revolution, but I'd place the PS2's exclusives above those. However, Gamecube did have the best versions of the new Sonic titles, despite the games themselves being rather weak. Beyond that, there are just multi-platform titles. Though in most cases the Gamecube versions beat the PS2 version and are equal to the Xbox version.
Then, with Xbox, when it came to exclusives you had Sega GT 2002, Crazy Taxi 3, Orta, JSRF, Gunvalkyrie, The House of the Dead III, Otogi & Otogi 2, Spikeout: Battle Street, and ToeJam & Earl III: Mission to Earth. Right there you have a strong number of titles, covering a wide range of genres. Then there were the multi-platform titles like Headhunter Redemption, Super Monkey Ball DX, and OutRun 2006: Coast 2 Coast as well as the many sports and Sonic titles. In many cases the multi-platform titles were the best or second best to the Gamecube version (though not nearly as far behind as the PS2 version).