Checkups keep you healthy indirectly.
If you tell someone "you have cancer but it's growth rate isn't fast enough yet for it to be worth treating" that will have a very different impact than "you're fine but we need to keep an eye on you to make sure you're in optimal health". But really they describe the same state of the world.
The alternative being proposed is to lump the people who are fine in with the people who are going to die unless they get treatment urgently and effectively forget or destroy knowledge about their state so that the amount of anxiety avoided in the first group exceeds the amount of horrible death and pain in the second.
It seems pretty clear that we have a ... problem ..(?) somewhere if we're even considering this.
I'm going to guess that the problem is that we can't deal with hazardous information very well. The knowledge that a person has a specific not-worth-treating cancer somewhere is hazardous to them and they need to be insulted from it. But it is not hazardous to the medical system, rather it is useful since with further monitoring it can turn into something actionable (a successful cancer treatment instead of patient death). But people are expected to be managers of their health (which makes sense), so that means they kind of have to know this hazardous information.