What are examples of how non-uniformity can be useful? I'm curious about ways in which you have seen non-uniformity be useful in computation. One way is randomness, as in $BPP \subseteq P/poly$, and another is look-up tables which are used to show that all languages have non-uniform circuits.

In particular, I'm interested in ways that objects known to exist via the probabilistic method and other non-constructive (or not-constructive-enough) proof methods can be leveraged using non-uniformity. I'd prefer the examples to be natural, not contrived. To be clear, a circuit for a contrived problem could be something like: given some language $L \in P$, I create a polynomial size circuit by computing some really hard function $f(|x|)$ using my advice and asking whether $f(|x|)^{n/|f(|x|)|} \oplus x \in L$.
An example I like is the argument that $\mathrm{NE\subseteq coNE}/(n+1)$ by counting strings in the language (see e.g. https://blog.computationalcomplexity.org/2004/01/little-theorem.html).