So... When I initially skimmed the article, I glanced right over the schematic showing the design, thinking it was an advertisement (correct, I guess?), hence some "reinventing the centrifuge" on my behalf in this thread.
Anyway, in the schematic, it does say "medium vacuum". The LHC is kept at "super duper vacuum", around 1E-9 Torr, to minimize collisions and keep the beam collimated. Let's say "medium" goes down to only about 1E-1 Torr. The delta between 760 Torr and 1E-9 vs. 1E-1 doesn't really matter, because from a structural consideration, you have a delta of just about 1 atmosphere in both cases. OK, so I know my little 1 meter diameter by 2 meters long cylindrical vacuum chamber (at ~1E-7 Torr, but irrelevant here) needs about 4 to 5 inches (~11.5 cm) of steel shell thickness to keep from deforming due to the delta between pressures over long periods of time. A relatively flat interface of pi*(50m)^2 area between "medium" vacuum and 1 atmosphere will require an absurd amount of material to prevent implosion. After considering it for while, I don't think building it underground gets you anywhere. Maybe if you hollowed out a cavity somewhere under 1 km of granite? With an investment cost large enough to do something like that, and considering how likely it is to eventually fail and destroy itself (see my comments above), this is still nothing but a fairy tale.
Even if miracle after miracle after miracle got SpinLaunch a functioning facility, I can guarantee you that the energy costs would make the traditional method of using rockets much, much more feasible. Approaching this from a carbon emissions standpoint might see the two methods a little closer, but now we're talking a 300+ man-hours case study, and I already have like two jobs.
note: too defeated for unicode today, bruv. I'm just glad that I made it back here to naysay some more :).