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It seems like when I run into this same scenario, I'm dealing with a mature URL that has simply positioned itself in a market and been able to hold. It's not that they are technically doing things "right", but apparently are not doing enough wrong to affect the gravity of their age and backlinks.

I think the original registration on that domain was 2003. Are there any other sites ranking around some of your key phrases that are less than 1-2 yrs registered? Age could be a factor.

Just to cover part: Never simply "oblige the request". That's bad for everyone. I always let a client tell me what they THIINK they need to rank for and how. Then we explain with data why that should or should not be part of the strategy.

We have clients that make recommendations for changes they want and they understand that if we say "it's because of SEO". They will back off and let us maintain the process.

Partial SEO would be, a home page optimization? That's fine as long as long as you verify and understand the scope of the rest of the site. Nothing wrong with "doing  a little work" on your highest possible ranking piece of real estate. In the end, it's all in the reporting and measurement. You just have to be able to communicate whhat you are actually going to deliver.

Hope this helps a bit.