or some similar kind of device that turns the momentum of electrons into light. I'm a little surprised that they didn't try something like a FEL first instead of that terribly problematic device that uses highly inefficient lasers to blow up tin droplets, itself a high-loss process that produces contamination and resulted in years of delay developing materials for
They tried both in the initial design phase, there's upsides and downsides, but ultimately thought that the tin droplet laser was more liekly to actually get done more or less on the time schedule requested, and so that's where the bulk of the capital went.
Interestingly China has been continuing working on the synchotron based EUV litho idea (in addition to work to create domestically built tin laser EUV lithos machines).
My bet is on plasma Wakefield accelerators to feed the FEL. But yeah a synchrotron might do as an intermediate step. Free Electron Lasers can be tuned to different wavelengths all the way to x-rays.