What you describe is still a problem with the institutions, because it is ultimately the institutions that provide the incentives (in the form of jobs). You're right that they're using bad metrics, but it is the institutions who are making those bad decisions based on the bad metrics.
There are lots of better things, like people making hiring and firing decisions based on their evaluation of the content of papers they have actually read, instead of just a number. If someone is publishing so many papers that a hiring committee can't even read a meaningful fraction of them, that should be a red flag in itself, rather than a green one.
It's true that hire and tenure decisions are under the institution's control. But a lot of funding comes from external sources, and most public funding uses some sort of publication-based metric. There are exceptions, but that's the game. The CV of your PhD's is often judged by the publication list and the corresponding citations. That's research institutes where they might go, other universities, large companies etc. will look at this. It's difficult to change this system as isolated player, and coordinates efforts so far failed on the "what else" question.
A problem with the public sector in this instance is it has money to spend, but no way of allocating it particularly well.
It will just pick the best allocation metric it has available, even if that metric would never stand up to scrutiny in the private sector, or any more directly measured domain, public or private.
I think the state could simply allocate money to long-lived scientific institutions and let the experts there handle things as long as there is no obvious corruption.
Self-regulation has a tendency to either work well for a few years, then gradually become corrupted... or be corrupt from the beginning.
A distressingly high percentage of humans like zero-sum status games. More people are happier when status is recognized as a semi-unbounded positive-sum game.
To dig even deeper into the problem: you have to get a large number of institutions to agree to stop this at once, none will voluntarily risk their (generally) working pipeline and system first. It disrupts a lot of different things and takes them out of the currently established model that everyone still uses to measure success. It reminds me of how most people who say “well not everyone should go to college!” Are obviously omitting “…except for my kids of course.” It borders on an expressive response, it’s not something anyone wants to actually take action on.
There’s not a whole lot to gain for the individual or even the institution unless they hit an absolute home run on the first try that also shows positive results very quickly. More than likely the decision will be questioned at every turn