Yes. The problem is that the one whose pocket the money comes from is not the one who selects the broker. Thus the person with the financial incentive to make that choice wisely is not the one making the choice. This is why it's a principle-agent problem.
The majority of apartments in certain cities charge this fee, so if you care to find one that doesn't you'd have to look for advertisements explicitly mentioning "no agent/broker fee". Maybe it's just my circle, but even the people I know that will look in less desirable locations for cheaper rent do not bother to look for this. It's also not far off from various psychological tricks seen in business - consumers do not behave rationally.
I'd argue it's even a bit worse than other sticker price bait and switch situations, because how many people know to look for this? Mostly only people that have signed a lease in one of these cities before, which for a place with a lot of transplants and subletting can be a surprisingly small number.
At some point I wouldn't even call it irrational to ignore the fee. Once you've decided on a place and receive the rental app to fill out, are you really going to turn back because you found out about the agent fee? Especially as time runs out to find a place and you realize more than half of them do the exact same thing?
I honestly wouldn't be surprised if some landlords have come to an agreement with the agent they use to get a bit of a kickback from the broker fee, unbeknownst to the buyer.
It works similarly when software is chosen by people who have to use it versus those who do not. IDEs and text editors are usually chosen by the users, so there is competition on usability between different options. Timesheet and other HR software are usually chosen by upper management, and the people actually using them cannot switch, so there is not the same kind of competition on usability. Instead, they compete on other things that make them more appealing to those who can make the decision.
Tenants can't "shop around" for a different letting agent that won't charge them £250 for a £50 credit check. Landlords can.