It all depends on the airline, the union, the contract, etc. Some airlines like Southwest the flight attendants bid a position A/B/C/D on board when they bid their work schedule for the month. So those who are senior and like riding in the back with the least to do are free to do so. A number of other carriers that have a lead or similar, many times there is a class or training to be certified but it's whoever is senior and qualified... they just have to log it as sometimes there's a pay difference of $1-3 an hour.

But yes, Purser role such as a premium market/cabin is mainly the one responsible for the aircraft paperwork (crew customs declaration, reconciling any onboard sales and deposit, duty free reports, etc). It may have some sort of qualification and likely some training but doesn't mean that person always flies Purser, or is very senior at all. An old joke about crew positions says that where you'll find the most senior flight attendant on an international flight is the one working galley where they don't have to face any passengers at all. Some more junior folks go for purser because they think it will, or does, yield "better" flying.

United has proposed a couple of times, including recently, wanting to make the Intl Purser a management position. This would give them some level of authority. Most union contracts prohibit one union member from directing the work of another or enforcing discipline. In 20 years in the airlines I've seen this applied from onboard situations where nobody is really in charge and someone just gets an extra $1 an hour to hand the paperwork to the gate agent on arrival to ramp positions where the workers were able to just pick what assignment they wanted off of a table because there wasn't a management supervisor present, so the person determining assignments can't assign them and just printed off the flights and set them out for people to fight amongst themselves over which one had the least work involved.