They are very wrong to make that claim. They have apparently lost touch with the vital understanding that evidence must be interpreted, and our starting assumptions guide our interpretations. I would never say that the current state of the evidence favors old earth, because both the young and old earth paradigms run into anomalies and problems with certain things they cannot explain. Both are a set of starting assumptions that guide how all the evidence is interpreted. Without God’s testimony as to what happened, we could have no certainty at all about the age of the earth, and old earthers are constantly guilty of overplaying their hand.
I’m not familiar with the strongest arguments tbh. All I currently know is evidence for young human origins: @swamidass geneological proposals and AIG’s Jeanson’s genetic work. But those don’t refer to animals, earth, or universe. What are your top #5?
Check this out:

Age of the earth
Earth is now generally accepted as 4.543 billion years old, but here are 101 evidences for a much younger age of the earth and the cosmos.