The only argument that document takes about it actually being better is that with some vaccines you need two shots instead of one. That's not compelling as being "better" at all. "Better" would imply that, once the complete course of vaccination is complete, the immunity is somehow more complete, stronger, or longer lasting.

Quite often what is "obvious" isn't true. Take some time and do some reading on this topic and you'll find that vaccines can provide equal or better immunity than a natural infection.

Your comment was indeed false, and many many studies have been done to compare natural vs vaccine immunity. Vaccination with a highly effective vaccine, e.g. Moderna or Phizer/BNT yield higher antibody response, and are more protective against variants in study after study.