Tyre tread itself is not as important on a bike than a car for grip if road riding. But if you can see threads exposed or even see the smart guard through cracks when pumped up to max, you ought to change ASAP. I’d be worried about tyre blowout.
I've had tyres fail by blowing out at the tread once or twice, in the 1970's. And had one dangerous blowout with a nearly new but cheap tyre that melted around the wire bead after excessive braking. Never (yet) had those problems with good quality modern tyres.
With modern tyres I have had several failures at the sidewall, generally with 16" tyres on my Bromptons after abusing them on survey work. So far these failures have been gradual, with a bulge forming that gives some warning before the tyre goes bang.
In my experience (again mainly with 16" tyres on survey work), kevlar beaded tyres are much less likely to fail
at the bead than wire beaded tyres. Failure higher up on the sidewall has generally been caused by abrasion or impacts on rocks/pothole edges. In the three 26" x 2" models I know about I would rate Marathon Mondial best for this, Dureme second and Supreme last.
For rolling resistance I rate these 3 tyres the other way round, with Supreme first and Marathon Mondial last, with not very much difference between Supreme and Dureme. This is relative, IMO Marathon Mondial still rolls significantly better than a (modern) 26" x 2" Marathon Plus, which I also have some experience of.
Based on the photos of Julio's tyre, I would fit new ones if planning a tour of several thousand kilometres. Supremes if mainly on tarmac, Marathon Mondial (because Duremes seem to be unavailable) if doing a significant amount of riding on rocky tracks. But I'd be quite happy using Julio's old tyres for utility riding or shorter tours.
I generally discard tyres when:
- the tread is worn bare,
- I start getting lots of punctures due to the tread having too many small cracks and holes or just being worn too thin
- the tread has a large cut or (more frequent) the sidewall is damaged (cut or ruptured or abraded).
When I wear out a rear tyre I generally fit the new tyre on the front and put the partly worn front tyre on the back wheel to finish it, as I reckon a rear blowout would be less risky than on the front. After 12000 kms I haven't yet reached that stage with my oldest Supremes.