If there were no religions and if it were not for the Bible saying so, is it possible that scientists might look more openly at the evidence, and have more freedom to theorise about the age of the earth? Would they hold just as strongly to the current estimated age, or be more tentative?
Thi si curiously paranoid. You seem to be fantasizing that scientists are in some sort of conspiracy to deny Biblical literalism.
The facts show the opposite. Remember that geology started in Christian countries, and the first geologists believed in a young earth and in Noah's flood. They reluctantly abandoned these ideas because the evidence was against their cherished YEC beliefs.
We might better ask: would geologists have considered a 6000 year-old earth at all if it was not for the Bible?
There are many methods to estimate the age of the earth, each one giving a different result, ranging from thousands to billions of years. If it were not for the Bible, would each method receive equal weighting, instead of most attention being paid to one method (radiometric)?
If it were not for the Bible,
no-one would spew out unscientific YEC gibberish. As you seem to admit in your post,
you wouldn't, either.
Be honest, if the Bible said that the Earth was four and a half billion years old, would you not take scientific confirmation of this fact as a confirmation of the divine authorship of the Bible, instead of talking rubbish about this subject?