Matthew, I believe you have stumbled upon one instance of why SEO is so confusing to many people. There is a lot of complexity and the answer varies based on the situation.
I believe you have taken the above quote from: http://support.google.com/webmasters/bin/answer.py?hl=en&answer=62399. I would always encourage you to share the source of any quote, so others may evaluate it within full context.
You shared you "found on numerous discussions and boards" information. Once again, I am confident you can find mis-information on any SEO related topic on "numerous boards". In 2011 when I was working with sites penalized for manipulative links, many sites had posts stating "Google will never penalize for manipulative links. They can't...." On our journey towards "SEO truth" please share the exact source.
Back to the point in hand, I would gently suggest you may be doing your clients a disservice by changing their geo-target. As it says in the same article from G, " ...if you have a site in French that you want users in France, Canada, and Mali to read -- we don't recommend that you use this tool to set France as a geographic target."
While some of your clients may only target local search results and therefore can be geo-targeted to the UK (a restaurant for example), you may service other clients who have a strong desire to rank in neighboring countries. Setting a geo-target for the UK may impact the site's rankings in other countries.
Why would you consider moving server locations? I am not passing judgment, but it sounds like you may be changing your client's server for your convenience, not theirs. My agency maintains a dedicated server in Florida, which works very well for US companies targeting a European audience. It would be more convenient for us to move our Canadian clients to our server, but it is not in the client's interest for us to make the move. Even if a client requested the change we would try to talk them out of it if the target audience was primarily Canadian.