What's your go-to technique for cleaning up background noise in a recording without losing the essence of the audio?

I use a variety of tools depending on the specific needs of each project. For precise spectral editing and surgical adjustments, I rely on iZotope RX 11. Waves Clarity and De-Verb plugins are excellent for overall removal and restoration, while synthetic restoration plugins are useful for rebuilding missing frequencies. Although these synthetic tools should be used cautiously, they have saved numerous projects for me, and when applied correctly, they can yield impressive results. Ultimately, each situation calls for a different approach, and the best solution depends on the specific problem.

Totally agree with the previous answer, RX (starting from version 8 ) is the best tool out there. And yes, the best tool is your hands and ears. I'd recommend though being accurate, when working on the vocal track in solo mode we tend to remove things that most likely won't be noticeable in the mix. 

Aside from the painstakingly tedious process of editing the audio regions themselves, I'd say the best tools out there would be the RX 11 suite from iZotope: 

https://www.izotope.com/en/products/rx.html

The way they read the audio signal to get rid of what you don't want whilst maintain a natural version of what you want to keep is pretty miraculous!