No, to put this more clearly, basically the undulators force the electrons which are of course speeding to follow a "sinusoidal" route, and thus this motion causes the electrons to emit light. SASE happens after this.
BTW, spatio-temporal splitting of FEL pulses can help mitigate the coherence and speckle-related obstacle and challenges we might
still face. Though, I wouldn't worry much about this yet.
Because we can't ignore FEL's characteristics akin to shorter pulse duration, narrower bandwidth, and of course higher repetition rate.
But last year, the researchers already submitted a journal related to testing done on High-power EUV free-electron laser for future lithography. And actually the good part is that despite the high peak power of FELs, the design of the scanner still ensures that the risk of damage to optics is minimal.
Paper which was published on 19 April 2023:
https://iopscience.iop.org/article/10.35848/1347-4065/acc18c/pdf