Hi Fergus,
There are some tests you can run on this official Google blog post:
Official Google Webmaster Central Blog: Using site speed in web search ranking
Whatever way people look at site speed, ultimately, you have to go with the one Google will see.
Page speed may be hosting related but from experience its mostly site or code specifc. Google Webmasters will give you an interesting graph that shows how it finds your site, and that ultimately is what your site will be judged on. But its also just a small signal. And your site has to be quite bad.
The webmaster tools will also tell you 1) what you ranked for and what the click-through-rate is and 2) where you've dropped. This can be quite a lot of data to go through but it is worth it.
The difference between hitting a peak and dropping are two different issues. Hitting a peak would probably mean that traffic is stable (with variance) and traffic going down would suggest something different.
I'd need to get a lot more information - traffic change by source, ranking changes, and so forth.
Whenever a company wants to change their website - we always ask why and what will you gain. For the most part, people want to update their website because they're sick of it - so the only thing they're fixing is their own feeling about updating a website. When we go through a managed process, the move to produce a new website drops to just 10% of instances. Lastly, we'd never be involved in a new site launch unless we fully understand what the placement of the current site is --- to minimise loss in traffic