Is counting natural? My experience tells me not! Many young children have to be taught to hear the pulse of music, and most of us have had to learn to keep a steady pulse. One needs to practise, sometimes, with a metronome.
My feeling is that before the age of the gramophone, musicians worried less about tempo, and went with the flow - one only has to listen to some 78s (transfers!) to hear that old way of playing. It's not that they couldn't play in time; and it is (usually) idiosyncracy: what they are doing is playing with flow. I like it!
Two tips for early grade aural tests
Here are a couple of anecdotal tips for early grade aural tests:
(1) I find that many young children can't clap in time, but they CAN tap their foot in time. Is that a common experience? Why would that be? Is foot-tapping entrainment, but clapping isn't - also, they try and clap like seals, rather than tap with the fingers of one hand into the other.
(2) Most young children can count 2-in-the-bar, but some seem to have difficulty with 3. Perhaps 3-in-the-bar is a more unusual time signature in these days when almost every pop song is in 4, or perhaps it has always been so. I have been known to tell ABRSM exam candidates, that if they aren't sure whether the piece in the Aural Tests in in 2-time or 3-time, that it is most likely in 3-time. I know that's bad of me, but it does usually work.
Notes: Entrainment on Wikipedia