The primary reason for thinking the earth is infinitely old is a metaphysical one, not a scientific one. It's the desire to avoid a beginning to the universe.
Indeed, many scientists initially opposed the theory of the "Big Bang" for just this reason. The theory entailed that the universe came into being out of nothing (ex nihilo) at some point in the finite past.
But why would scientists object to a cosmic beginning? The answer is that once we start talking about an origin of space-time the next step is to posit a cause of that origin and that soon starts to sound a lot like Genesis 1:1, which is a horrifying prospect to naturalist scientists and philosophers.
There's a principle most philosophers accept called the Principle of Sufficient Reason (PSR). This principle states that every event must have a cause adequate to account for the event. If this is so then a universe that had a beginning must have had a cause adequate to account for the universe we see.
A cause of the universe would have to be extraordinarily powerful and intelligent. It would have to transcend space and time (which are part of the fabric of the cosmos), and, since the universe has generated personal beings like ourselves, it's reasonable to assume that the cause of personal beings is itself personal.
Of course one could deny all this by denying the PSR, but that seems a pretty steep intellectual price to pay to avoid having to acknowledge that there is a Creator.
It would, in effect, seriously cripple, if not altogether destroy, science, which is an enterprise largely based upon the PSR.
