...This sounds vaguely worrying.
In what way?
Well, how do they choose? What if they choose someone who is wildly unsuited to the role? Like Frieza, for instance?
How do they choose? The only thing we have seen about this is when Whis the offered the job to Goku out of the blue.
Depends on what you mean by unsuited, Beerus is very well mannered but he's more volatile then nitro-glycerine. He would destroy a planet just because someone was rude to him. There are only two things in the universe he isen't allowed to destroy as far as we know, the sacred world of the kais and his temple. If we wanted to destroy half a galaxy the Kais wouldn't be able to stop him because that's his job. They have no authority over him and he may do as he pleases. He could even destroy a Supreme Kai or even the Grand Supreme Kai himself and he would not be rebuked .
That being said there are things in place to keep the God of Destruction in check. Firstly he spends most of his time sleeping, sometimes for hundreds of year. This seems to be compulsory. Secondly unlike Supreme Kais who can travel anywhere in the universe at will the God of Destruction is grounded. He can only travel as a mortal would (using his ki to fly and move at super speed). If he wants to travel across the universe quickly or to Other World he needs assistance. He also doesn't have the Kais Far-Seeing Arts, a technique which allows them to see what's happening in the universe from afar. This means that the God of Destruction is blind to events taking place in the universe, however while sleeping he can sometimes have prophetic dreams (although Beerus's almost never come true). Also while they have no authority over the God of Destruction once every 1,000 years, the Supreme Kais and the God of Destruction must meet with each other and hold a coordination meeting. Interestingly the God of Destruction dosen't just become all powerful, their power grows over time after becoming a god. For example its been stated that Beerus was the one who sealed away Old Kai in the z sword (this happened they got into an argument over some trifling thing during one of their meetings and the short-tempered Beerus sealed him away because he wasen't allowed to destroy the sacred world of the kais). At the time he was not as powerful as Super Buu but over the years he become so strong that even Vegito wouldn't be match for him
The God of Destruction also seems to have a symbiotic relationship with his Attendant. While he spends most of his time sleeping, his Attendant never sleeps and constantly monitors the universe so he can fill the God of Destruction in on what's been happening while he was asleep. The Attendant also acts as transport for the God of Destruction and will take him where ever he wants to go, however their form of travel is different to the teleportation of the Supreme Kais
The Attendant to God of Destruction apprise to be from the same race as the Kais and Supreme Kais but he is for more powerful than even Grand Supreme Kai. Not only does he care for and assist the God of Destruction he also acts as the Gods martial arts teacher. It is also his job to discipline the God of Destruction if he gets to wild.
As it stands Whis is the strongest being in the Dragon Ball universe