Hi there
Well yes, a CMS can have an effect on SEO performance. If the CMS makes elements on the page uncrawlable, or the website load very slowly, this could have an adverse effect on your performance. Similarly, if the CMS produces a lot of duplicate webpages, URL or content, which can often be a problem with e-commerce sites, that could lead to algorithmic Panda penalties.
However, looking at your homepage, the Googlebot sees this on the page - so it looks as though everything is fine with its crawl there. In addition, your site load speed appears to be good. I also can't find any obvious duplicate content or URL issues - the search function works well and each product listing looks to have good unique content, with no duplications of the pages (at least at very first glance).
So your CMS, at first glance, looks to be working really well. The issue that I see is that, as a result of your bad backlink cleanup, you have a very small amount of links pointing to your domain. Majestic SEO is reporting that you have gone from a peak of 9000 backlinks and 1000 root domains to just 69 backlinks and 25 root domains. That is a huge reduction in links. While being completely necessary, it essentially means that your off-site SEO is back to zero and so you need to earn more links. Now, it's about quality rather than quantity, but to give you a quick example: the top 5 results for "specialist paints" each have over 10,000 links to their domain.
You've done a great job with your onsite optimisation and an extremely thorough link removal process. What you're left with, however, is a site with very few links and so not a lot of power to help it increase its visibility in organic search. I would say that you need to replace the bad links you have had removed with strong, contextual and high-quality links in order to start performing well again.
I hope this helps.