This comment has stuck out to me for a while, and I thought I would ask about it if you don't mind.
We recently performed what I would consider extensive keyword research for a medium sized business, and they did not seem impressed with the keyword seed lists we assembled for them. To gather the seed lists for each page, we did two things:
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Head into their GA and GSC to see every single keyword people used to get to their website last year, and took each of those, then split them up into ideas for each page. We then did as many variations on them as we could think of by adding their city (San Jose), state, country, pluralization, etc., and wound up with thousands of keywords in their "our ideas" seed list.
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Second, we used Moz Pro Keyword Explorer for additional ideas, entering in their pages one by one, to see what Moz came up with, then exported those for each page as additional ideas.
Afterwards, we took ALL of our ideas and ALL of the Moz ideas, had them split up page by page (of course), and checked for the best combinations of volume, difficulty, ctr, priority, etc..
All that said, I believed this to be a rock solid approach until I saw your comment, which made me wonder. Do you feel this was a solid effort? Have you seen better results from a different approach? (In my mind, this was far better than what I believed what most other agencies would have provided them with, and I am almost wondering if their complaining is an attempt to renegotiate on their invoice -- something their owner is unfortunately keen on doing.)
Greatly appreciated,
Zack