Hi Eric,
First, I need to make a small correction to my earlier statement. I meant to state that duplicating text across PAGES would = duplicate content. Footer content is excluded from this. Most websites have the same content in their footer from page to page. Sorry if what I wrote was misleading. So, duplicate content isn't the issue here, but...
While I agree that your devil's advocate argument contains horse sense in some ways, the footer is probably one of the most obvious places Google would look for spam. If Google goes out of their way to state that they don't want blocks of city names on a website, I think it's wise to take this at face value. And I think that, because footers are historic candidates for spamming, Google would be especially down on this type of content being placed there.
I may be a little bewildered by your client's goals. If they don't want to rank for their service cities, why include them anywhere on the website? And if the goal is to let visitors know the areas they serve, why not at least put that on the homepage of the website, shine a spotlight on it so it's totally clear? Make a custom map. Write a full description. Create unique pages and link to them from a top level menu.
Relegating this vital information to the footer, which many people won't even see, just doesn't make sense to me. The areas an SAB serves are critical data and should be highlighted in every way possible. When I develop websites for SABs, I create two main types of pages beyond the typical homepage, contact, about, etc.:
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Service description pages (one for each service)
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City landing pages (one for each major city served)
To me, these are the basics of any local business website, and I would be puzzled by a client who didn't 'get' this. I work primarily with small-to-medium local businesses, and I totally understand budget constraints, This is why creating a protracted timeline for getting all the necessary pages developed is often a good solution. To me, from what you've written, it sounds like your client may not have clear goals in mind in regards to how their website will serve them and their customers. Hopefully, as their marketer, you can help them create a plan that will take their website forward to becoming a true sales-generating asset. I hope these thoughts are helpful.