Now, beyond the question of an SPD candidate in 2002, there is a huge question lurking behind this entire complex, and I'm sure it's been discussed before here, but I'll address it nevertheless:
The Schröder era of 1998 to 2005 changed the German political landscape - and beyond it, society and economy, too - massively.
An SPD-led government cutting back welfare so severely as in the Agenda 2010 reforms, especially creating such a precarious low-wage sector, destroyed the SPD as the party of the working person forever. Admittedly, policies like raising the pension age to 67 and the VAT from 16 to 19 % only came in the great coalition 2005-2009, but by then the SPD was hopelessly on its odyssey into neoliberalism. Whatever the SPD has done afterwards, be it minimum wage laws or the current Bürgergeld law, they will never recover. Into that political void, a new polical party (Linkspartei, then Die Linke) stepped, the synthesis of the post-communist SED with the Western anti-Agenda reform protest group WASG. The strength of Die Linke in the late 2000s and early 2010s changed a lot by itself: by now, they no longer function as a post-communist representative of the East and also not as a populist projection space.
Also, participating in Schröder's cabinets changed the Greens deeply. They participated in the Kosovo War, in KFOR and ISAF, and thus severed their pacifist ties. Without that, not only would they now be the party most staunchly supporting military support for Ukraine right now. Without this demonstration of pragmatism, the Greens would probably not have been able to break into centrist electorates which they have repeatedly been able to capture ever since, going so far as to achieve more than 30 % in Baden-Württemberg and having a Green PM there. The Greens have finished their transformation from a leftist movement party into a progressive-centrist party elected by liberal urban middle classes broadly, but that all depended on their coalition action in 1998-2005. They did achieve something back then: the EEG (renewable energy laws) began the expansion of renewables in Germany and even created an industry there (which was ruined on purpose by later governments), and exiting from nuclear power was decided for the first time.
Now, these two transformations have complemented each other: as the SPD lost their hold on the centre-left electorate (in a more traditonal alliance between organised labour and progressive middle classes), the Greens have been able to capture a sizable chunk of it (and Die Linke in the meantime, too, but they lost it again). The Greens, much more than the SPD before them at least in the last decades of the 20th century, stand for cultural transformations among the middle classes, too. They not only snatched progressive voters from the SPD, but also centrist urban middle class voters from the CDU. That, in turn, has enticed Merkel to lead the CDU more into the centre of the political spectrum.
So, the big question is:
Had Schröder not won in 1998, would all these transformations not have happened? THat is, would Germany now look more like its Scandinavian neighbors: with a Social Democratic Party still dominating the centre-left, and Greens being a decidedly leftist party but small, while the CDU would have stayed more conservative and never lost hold over the radical fringe of conservatism? Would the PDS have stayed the post-communist party of the East, and of a few radicals in the West?
Would that even have meant that the neoliberal reforms of the early 2000s would not have become a reality, probably only partly advocating for by the CDU but still opposed by the parties of the centre-left?
My answer to this question has changed over time. I was one of those red-green voters who were really, really upset with the Agenda 2010 and who voted Die Linke for quite a while, even radicalising myself ideologically. I tended to blame Schröder (and much of the rest of both governing parties) for all the neoliberal stuff that they did, and that was done afterwards, too, or at least for having started it, and I thought that if Schröder had not opened it, the neoliberal Box of Pandora would have remained shut and Germany would not have the kind of unfettered low-income exploitative segment of its job market.
Looking at it with some historical perspective now, I think that's not what would have happened. The political discourse 25, 20, even 15 years ago was so much more economically neoliberal than today's that some neoliberal reforms would have been pushed through by whatever government, even another SPD-led one. Something like what happened over the past weeks in Britain - Truss and Kwarteng announcing tax cuts for the wealthy, and the financial markets plummeting in London, then conservatives backpedalling on it all and even Truss eating her own agenda - would never have happened back then, neither in Britain nor in Germany. It shows how much the political discourse has moved away from neoliberalism. But that's because we've learned what its effects are. Before that, neoliberal promises were strong.
And beyond that, the SPD's ties to the working class vote were bound to erode at some point anyway, if you look at long-term trends in unionisation and all that. It might not have been quite the Pasokisation they went through IOTL, but the 40%+x results of the 1990s were a last hooray.
And the Greens were on their path towards pragmatism anyway. Their breakthrough might have been limited or postponed by another Kohl victory, and ecological transformations might also have been delayed. And the PDS could probably not have played the Eastern card forever, either, although on that one I'm not fully sure.
So - even without Schröder, 1998 (or some similar year) was really likely to be the starting point of a political realignment, and reforms were very likely to happen, and the outcome they would have on the political spectrum, while not determined to be exactly what we see today, would still have worked in directions that we would recognise. Yes, Schröder's victory in 1998 was a watershed, but it would have been quite an implausible TL to imagine a Germany in which none of that for which Schröder's years stand happens and Germany somehow remains in the political alignments of the late 20th century for two or three more decades...
. At and after the party rally for nomination he would very likely framed his retreat from his former statement as "serving the party's will and 'plea' of the party to serve as candidate". ... or some other shennanigan as excuse