Seriously, why did SEGA games begin to suck almost immediatly after they left hardware?
Two things.
First off...they didn't. After the Dreamcast died Sega soon released Shinobi, Phantasy Star Online Episodes I & II, Gunvalkyrie, Shenmue II, Jet Set Radio Future, House of the Dead III, Super Monkey Ball, Virtua Fighter 4, Toe Jam & Earl III, Panzer Dragoon Orta, Sega GT 2002, and Billy Hatcher. These games seem to be in line with the Dreamcast philosophy and quality. This is why the Xbox was commonly called "The Dreamcast 2" back in the day as it had so many titles that would have previously been exclusive to the Dreamcast had it not died. The problem was the exact problem that plagued Sega with the Dreamcast, most of these games just didn't appeal to the market at the time. I mean yeah Shinobi and Jet Set Radio Future are great games and all, but at the time when people were salivating on the cinematic nature of Metal Gear Solid 2 and the "holy shit I can do whatever I want!" in Grand Theft Auto III, it didn't really appeal to people. As most of these games underperformed, Sega tried to figure out just how to make their games profitable which led to them taking a different route from their previous philosophy. If you want to be technical about it, the Sega that we once knew didn't really die until late 2003 after the release of Outrun 2.
The second reason is, as you likely guessed, restructuring. Sega wasn't making that much money after leaving the console business. Because at the end of the day games like Panzer Dragoon Orta aren't going to be multi-million sellers. The post-console enthusiasm soon went away and Sega slowed down their "Sega console games for everyone!" mentality. But things really went south when Sega became owned by Sammy. Sammy pretty much constructed Sega of Japan to be mostly arcade focus, and when I mean arcade focus I mean mostly on super safe sports titles at best to pachinko machines at worst. Prime example were Smilebit and Overworks. In case you aren't familiar with them, Smilebit developed Jet Set Radio, Jet Set Radio Future, and Panzer Dragoon Orta. Overworks developed Skies of Arcadia, Shinobi, and House of the Dead III. Again not the best selling games (besides Shinobi which wasn't far from being a million seller) but were great in quality and were clearly responsible for Sega's image. As a result Sega tried to make these studios profitable.As a result, quality suffered. For example due to the changes Smiliebit began to make nothing but soccer games and and horse derby racing games. That's right they went from working on Panzer Dragoon to working on horse derby games... While Overworks...didn't really make anything again (I'll get to that in a minute). Sammy's chairman became Sega's CEO in August of 2003, which likely had a huge influence on what games the company released. If you remember, 2003 was the last year Sega consistently released their "Sega-like" trademark games.
But if you want to find an exact date of when things went to shit, it was July 1st, 2004. That is the day the Sega of the old truly "died". Sega was brought and absorbed into Sammy and as a result many of Sega's best developers including Smilebit, Amusement Vision, and Sega Rosso were dismantled. This also included Overworks who seemed to stop making games a few months earlier. I guess once Sega was headed by Sammy, they decided to just have some of these studios to stop developing games to prepare for the inevitable merger.
And that's more or less it. It's pretty sad because 2003 was the last year Sega that we knew existed, and that was the year the company FINALLY posted a (albeit tiny) profit. Possibly if the company held on things could have turned out different, but I doubt it. They just didn't have games that could have been massive sellers.