What are the hazards and risks if I keep using a worn rear tyre? I found similar questions:

What are the risks of fenders/mudguards?
What are the risks of untrued wheels?

But I haven't found a structured analysis on the subject of tyres.
Stack exchange technicalities: It is the rear tyres on two bikes in my family that seem to be a concern, so I kept my question specific to rear tyres. But please feel free to edit the title and question if you feel that provides a better balance of things like keeping the question specific vs effort required to provide a clear answer. I just don't know if it's better to deal with just the rear by itself or both wheels together to make this question useful.
Logic: Ideally I would love a structured answer, perhaps based on this article about hazards vs risks. Naturally there may be other ways to approach this.
The hazards that come to my mind are:

Loss of grip in a bend/obstruction avoidance
Braking performance - this is probably fairly gradual until the tyre punctures?
Puncturing - the resistance can probably be measured, with some graphs based on some standard puncturing object vs tread depth?

How the hazards translate to risks then probably depends on several factors, like the sort of riding you do, what mode of failure is developing on the tyre (cracks vs thinning) etc. You could probably write a books about this, right?
Clearly if I'm already pushing my tyres to their limits in bends/braking etc. then the tyres need to be it tip-top condition. I don't think I ever get anywhere near these limits, but I guess I am conscious about emergency braking in commuting on roads.
Do countries have government rules for bike tyre safety, like the MOT rules for cars in the UK? I'm not asking about the annual testing and certification, I'm on about the minimum standards of roadworthiness.
I really thought that this subject has been dealt with on this site or elsewhere, but I couldn't find anything complete/clear/structured.
It if matters, the reason I'm asking is I am a bit of a pessimist (perhaps a primary characteristic), and therefore both a cheapskate and safety-conscious, which I would argue are both secondary characteristics on one hand, but clearly conflicting on the other. Safety and frugal are valid tags on this site, pessimism isn't:-)
If it’s just the rubber tread which is worn down and thinner: Keep using it. With treadless tires you’ll only suffer from increased susceptibility to punctures. Of course with knobby/treaded tires your grip on soft surfaces will gradually decrease.

If the tire’s structural integrity could be compromised: Don’t use it. It could fail catastrophically at any time. Things like bulges, abrasions of the sidewalls, cuts or large cracks are all bad signs.