Why are lacunary series so badly behaved?

You can read more about this in the excellent survey

J.-P. Kahane: A century of interplay between Taylor series, Fourier series and Brownian motion, Bull. London Math. Soc. 29(1997), 257-279

In particular you can learn from this survey that the phenomenon you mentioned is rather typical. It's definitely worth having a look at it.

The mentioned gap theorem was generalized by Fabry (Acta Math. 1899, pp. 65-87): if the power series $f(z)=\sum_n a_n z^{\lambda_n}$ has radius of convergence $1$, and the exponents $\lambda_n\in\mathbb{N}$ satisfy $\lambda_n/n\to\infty$, then the unit circle is a natural boundary for $f(z)$.

Turán (Acta Math. Hung. 1947, pp. 21-29) gave a simple proof which might provide some insight into the phenomenon. His main inequality, from which he deduces the result, reads as follows:

$$ \max_{0\leq x\leq 2\pi}\ \left| \sum_{n=1}^N a_n e^{i\lambda_n x} \right| \leq \left(\frac{48\pi}{\delta}\right)^N \max_{a\leq x\leq a+\delta}\ \left| \sum_{n=1}^N a_n e^{i\lambda_n x} \right| $$

In other words, the key feature seams to be that on every arc of the unit circle, the partial sums are considerably bounded away from zero. For more details I would recommend to study Turán's paper.

Maybe your question is backwards. Natural boundary at the radius of convergence is the usual thing, and analytic continuation outside the circle of convergence is the fluke. Only VERY SPECIAL series have continuations.