I don't know if they use manual welding or robotic welding. But robotic welding is well established and is justified for the volumes of work that SpaceX carries out. What is more difficult is to avoid vertical weld seams on its cylindrical segments. I'm yet to encounter a roll forged cylinder that big, especially with stainless steel. (Disclaimer: I have no direct experience with industrial metalworking)

> If any failed weld can lead to a catastrophe like this, how would you guarantee the quality of each weld without going into nuclear power plant construction level of costs?

That is done using Non-destructive testing (NDTs). The usual methods are high-energy X-ray imaging, ultrasound testing (UT) and dye penetration testing (DP). These methods are usually reliable in catching such faults - even for machinery that's in use. For example, turbine blades in a jet engine.

Updated: As the other commenter pointed out, robotic welding doesn't ensure elimination of defects. The robotic process is more consistent and therefore reduces the defects. But uncontrollable random variations can still cause defects that could fail later. The only way to eliminate them (almost) entirely is to identify them with NDT and rectify them as long as the defects are within a certain tolerance limit.