Vestigiality is not about having no function, but about losing the original function. Vestigial organs are also often in an atrophied state when compared to analogs in other species. The appendix is indeed vestigial, and while not entirely without function, it is almost entirely without one, to the point that you can not have it and not even realize it.

>When the most powerful telescope could see N lightyears, people believed the universe was N years old. [...]

What are you talking about? Before the 20th century it was believed that the universe was eternal. After then, advances in determining the age came primarily from theoretical models, not improvements in equipment.

>Yet astronomers and cosmologists refuse to learn the lesson, and keep repeating the same mistake.

You seem to be under the misapprehension that the goal of science is to not make incorrect statements. It's not. It's to incrementally (by necessity) learn about reality. There's nothing with the statement "by our best current measurements, the universe is about 13 billions years old" even if tomorrow new findings point to it being twice as old. To demand otherwise would mean that no conclusions can ever be drawn, because necessarily all scientific conclusions are tentative. It is true at all times that tomorrow's evidence may overturn today's conclusions.