Suppose in an NFA we have an $\varepsilon$-move from a state $q_0$ to $q_1$. According to Sipser,
Without reading any input, the machine splits into multiple copies one following each of the exiting $\varepsilon$-arrows and one staying at the current state. Then the machine proceeds nondeterministically as before.
Here is an application of this logic to our case: upon reaching the state $q_0$ our machine splits into two copies: one in state $q_1$, one in state $q_0$. For the latter copy, execution continues as before, so again another copy of the machine is created in state $q_0$. Thus, a copy of the machine always in state $q_0$ will exist indefinitely.
Is this true? If no, why not? How should I then interpret the description given above?