It's not false advertising to claim your food is the best because everyone literally does that. It is just a slogan to try and sell more pizza. Nobody actually believes whoever claims to have the best of anything actually has the best. They know that's just how the advertising and marketing world works.
Yes, companies engage in too many frivolous lawsuits because it shows their power when they try to provoke another company. They will go to court for no reason at all in order to scare another company into conceding something. In this case, they simply wanted to hurt the other company's pizza sales.
Companies have the right to protect their intellectual property, but it is not the role of companies to police other companies, nor should companies sue just to gain a competitive advantage. In the case of Pizza Hut suing Papa Johns, even if Pizza Hut believed that Papa Johns was lying, it was not Pizza Hut's place to sue. Clearly Pizza Hut was only trying to prevent the success of a competitor, which should be seen as bad business.
Papa John's slogan "Better ingredients. Better Pizza." is just an expression of opinion, or at worst marketing hyperbole. For Pizza Hut to sue on the grounds of false advertising is a waste of time and money. Pizza Hut should use its money to improve the taste of its own pizzas and advertise the fact rather than waste that money on overpaid lawyers.