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Depends on what type of sites those "not so great sites" are.  If they are viewed as spammy, no they won't help.

I agree with your SEO that text contained at the top of the article is given more weight then text at the bottom of an article. It is really difficult to provide any estimates on how much more weight. If Google would allow us to look at their algorithms, we would be able to provide a specific answer. Since that is not the case, we are left to making our best guesses using experience and related knowledge as a guide. We know that keywords in a title have more weight when they are earlier in the title. We know it makes sense from a user experience perspective to offer your best, most relevant content upfront rather then buried at the bottom of an article. I would build upon that logic and my experience to suggest that relevant text be provided at the top of the page (all things being equal).

You can run a graphical report showing your Ranking History for each keyword targeted since you started your campaign. To do this log into your campaign.  Click View This Campaign/All Keyword Rankings. On your Rankings Report click on the ranking you want to see charted.  For instance, if you rank #1 on Google for a keyword this week, you can click on that ranking and a graphical report showing your Ranking History for that keyword since you started your campaign for that word will appear.

Are the keyword terms this "SEO Company" is ranking you under receiving any actual searches?