The book Introduction to Banach spaces II by Daniel Li and Hervé Queffélec has a chapter on "the method of selectors" which seems to be a subgenre of the probabilistic method. Here is a sample of the applications given in the chapter:

• A theorem of Bourgain allowing on the extraction of "big" quasi-independent sets, with applications to theorems of Pisier and Drury about Sidon sets.

• A theorem of Bourgain on sums of sines: for any integer $N \geq 1$, there exists a subset $\Lambda \subseteq \mathbb N^*$, of cardinality $\lvert \Lambda \rvert = N$, such that $\left\| \sum_{k\in\Lambda} \sin(kx)\right\|_\infty \leq C_0\,N^{2/3}$, where $C_0$ is a numerical constant. This has an application to vectorial Hilbert tranforms.

• A theorem of Bourgain on the geometry of Banach spaces (a lower bound on the "K-convexity constant").

I don't know these subjects too well, but I feel that each of these examples live in a rich ecosystem of results, some of which also answer your question. For example, here are two other results on trigonometric polynomials which can also be obtained by probabilistic methods:

• Salem and Zygmund proved in 1954 that given amplitudes $\rho_1, \ldots, \rho_N$ and phases $\phi_1, \ldots, \phi_N$, one can find (random) signs such that $\left\| \sum_{k=1}^N \pm \rho_k \,\cos(kt+\phi_k) \right\|_\infty \leq C\,\sqrt{\sum_{k=1}^n \rho_k^2} \sqrt{\ln N}$, where $C$ is an absolute constant.

• Uchiyama proved in 1965 that there are subsets $\Lambda \subseteq [\![ 1, N ]\!]$ such that $\left\| \sum_{k \in \Lambda} e^{ikx} \right\|_1 \geq c\,\sqrt N$, where $c$ is an absolute constant.

I know I would love to see a user-friendly survey on this kind of constructions. Kahane's book is probably very enlightening, but it remains a bit formidable-looking to me.