No anxiety required.
In multiple trips to the US with my dog, connecting in various gateway cities (MIA, JFK, DFW, PHL, LAX, DTW, DFW...) I have never been subjected to an animal inspection. Most of the time, the immigrations/customs officers didn't bat an eye, and the two times where I was asked about the animal a quick check of her pet passport where the rabies shot was documented was all that was required.
I had USDA inspect her (kibble-type) food I brought with me a few times, though and that took a few minutes. So if you want to avoid being held up by the agricultural inspection, don't bring any (meat-based) food.
And even when you miss that flight and get put on the next one - that is no different with a pet than without one.