A COVID-19 booster is given when a person has completed their vaccine series, and protection against the virus has decreased over time.
Yes, it has decreased over one. No one is saying you have no immunity after 5 months. It may wane a bit, but you still have immunity. Delta is rather virulent though.
Nothing beats natural immunity, those that have had covid and recovered , their immunity lasts way longer than any jab or booster. Those that have had 3 jabs will suffer side effects.