No, the trusted role falls under the category of "Optional Server Settings", this means the bot will run perfectly fine without having the trusted role set.
However, the trusted role system is a feature if servers need to limit which users should have their birthday celebrated. A good example of the trusted role system in use was our Discord support server. In our support server, we had our trusted role set as the Staff role so only our staff had their birthdays celebrated in the support server.
The trusted role is the role that can be set up so a user needs it in order to have their birthday celebrated. If the setting
Trusted Prevents Message is enabled/true, then only users with the trusted role will receive a birthday message. Likewise, if the setting
Trusted Prevents Role is enabled/true, then only users with the trusted role will receive the birthday role. Note: if the trusted role itself is not set, none of these settings will apply, and anyone with a birthday set will receive both a birthday message and the birthday role (assuming both are set).
As explained above, the trusted role(s) decides whose birthdays will be celebrated in a Discord. All servers are allowed to set a trusted role; however, Birthday Bot Premium servers are allowed to set multiple trusted roles (up to 250!). If multiple trusted roles are set, the require all trusted roles setting decides if a member needs all of the trusted roles to have their birthday celebrated or, if the setting is false, the member only needs one of the trusted roles to have their birthday celebrated.