No, chaos does not ensure normal diffusion.
The typical example for anomalous diffusion are systems whose behavior alternates between jumps/bursts and relatively well-behaved regimes.
A prototypical case is indeed intermittency (see this introduction), but also prominent are chaotic Hamiltonian systems which are not fully chaotic, i.e., with mixed phase spaces – here the chaotic behavior is interrupted by long periods of almost-regular movement caused by the trajectories getting trapped around regular islands (as mentioned in this question and many other references).
This intermittent dynamics is often described by means of Lévy flights, which are random walks where long steps are more likely than for the Brownian motion. The step length probability distribution for these systems is then said to be fat tailed – which confirms your impression that “sufficient ‘randomness’ along the trajectory of a particle leads to a vanishing probability of ballistic flights”, only that chaos alone (being, after all, deterministic) is not necessarily sufficiently “random” to preclude fat tails. The chaotic pendulum, for instance, can be mimicked by
a stochastic model which [alternates] ballistic flights and random diffusion.
The link of anomalous diffusion with Lévy flights, its connection to other phenomena such as ergodicity breaking and the possibility of it being a transient regime for some systems is well articulated in this paper’s introduction:
Their behavior is dominated by large and rare fluctuations that are described by nonexponential decay laws, commonly referred to as Lévy statistics. Other characteristic features of these systems are ergodicity breaking, that is, non-equivalence between time averages and the corresponding ensemble averages, as well as aging, i.e. a manifest dependence of physical observable on the time span between initialisation of the system and the start of the measurement. Both ergodicity breaking and aging are in essence two sides of the same coin as they are intimately connected to non-stationarity or ultra-slow relaxation. These systems exhibit various forms of diffusion anomalies which were also demonstrated in numerous experiments. This kind of dynamics may not survive until the asymptotic long time regime, nonetheless, lately even its transient nature has been predicted theoretically and observed experimentally.
Thus, considering the examples given, chaos might not ensure normal diffusion, but strong chaos typically probably does, at least in the asymptotic regime.
