"It can't just be a case of "throw 100 different things at the wall and see what sticks"?"
This is exactly my point. You are rolling the dice by throwing one, two, five or 100 things at the wall and seeing what sticks. An audit examines the high level consideration points (my audits look at over 90 high level factors) then how they relate to each other.
In that process, a skilled auditor can identify issues that have more of an impact than others. The end result is a focused list of specific tasks prioritized for maximum long term value.
While I happen to be someone who specializes in audits, this is not why I advocate them. I do so because you specifically should not have to continually throw things at the wall, praying to the Google gods in hopes that something will stick. It's extremely inefficient and frustrating. Just as trying to evaluate a single competitor for one or a few phrases. In that scenario, you are continually having to play leap-frog with them and others.
In a proper audit scenario, with long term across-the-board sustainability in mind, it's not as much about what one specific competitor is doing. It's about getting a much more vast range of value on a much stronger foundation.
And too just because I do audits for a living I don't post this info here to get new business. I come here to help others grasp concepts most people fail to understand or consider. Many people can do audits - some even for free. The key though is an audit becomes a long term road-map. It becomes invaluable in your arsenal and helps cut through the online noise.
When an audit is done properly, it does not focus on any one set or sub-set of phrases - it's a holistic evaluation of the site's critical considerations. Sure, you can start working on aspects of the site specific to one set or sub-set of topics when you do the implementation work. However it's almost always a case where there are site-wide issues that are an underlying cause to a site's problems.