Thanks for raising this interesting topic, Valerie. I hope you and the little one are doing well!
The failings of the Newtonian model gradually became apparent in the late 19th and early 20th centuries. The stage was set for Einstein’s general relativity, with which everyone in this thread is at least familiar, I’m sure.
Before general relativity was even introduced, however, the age of the universe was known to be at least tens of thousands of years old. In 1676, hundreds of years before Einstein, Ole Christensen Romer estimated the speed of light with an error of only about 20%. In 1729, James Bradley estimated the speed of light with an error of only 1.5%.
Given the speed of light and the distance that light has traveled, you can calculate when the light was emitted from its source. Well before the advent of general relativity, astronomers knew that the Milky Way has a span of at least 10s of thousands of light-years. This observation set a lower bound on the age of the universe at 10s of thousands of years.
Today, we have much better instruments and a much more accurate understanding of light travel in the universe. The Hubble is viewing electromagnetic radiation today that began its journey billions of years ago.
In the late 19th century, geophysicist John Perry estimated the age of the earth at 2 to 3 billion years based on heat dissipation equations. This sets a much larger lower bound on the age of the earth.
The next big cosmological theory will be fascinating, whatever it may be. It would be quite mistaken, in my opinion, to think that somehow the new theory will shrink the age of the universe from billions of years to mere thousands. The evidence for billions of years is overwhelmingly strong.
Hope this was helpful. Keep loving your little one!