Hello mrlolalot (nice name, lol!),
As you already mentioned, Stores Direct has suggested what is about the best way to handle this, which is to not link the short description text, but to use it to improve usability. To that I would add, try to work in a few keyword phrases that are optimized for the page on which the short descriptions appear, which shouldn't be difficult since they are appearing on a category page that is relevant to the type of product they describe. I would also mention one potential downside of showing short descriptions on category pages is that products that live in multiple categories could have their short description showing on more than one page. Still, I think the reward is bigger than the risk as long as you don't have the same products in more than a couple categories. If most of your products are in four, five, six... categories than you're going to have a pretty big duplicate content problem on your hands.
Also as you already mentioned, linking the short description isn't going to do you much good since you're linking the product title above it, which is probably a far-better link to have anyway, keyword-wise. Linking the title of a product is the epitome or relevant anchor text. And I'd stay away from nofollow linking any internal page on the site with the exception of perhaps cart and account type pages, which should be noindexed in the meta tag and/or disallowed in the robots.txt file anyway.
But to answer your question: Yes, it can be a bad thing to display non-linked short product descriptions on the home page and category pages IF you aren't using the keywords for which you want the home and category pages to rank; and IF you are showing that same short description on several pages (i.e. multiple categories and the home page). But I do like short product descriptions to appear on category pages for reasons Stores Direct explained in his first answer. So my advice would be to adjust the keyword use if you can, while addressing the duplicate content problem if that is an issue.
Sometimes with these established sites you'll find the category pages already ranking for the product-level keywords and you may worry that changing the strategy will cause the page to no longer rank. An example:
Your widget category might rank for Small Purple Widget because you have a short description on there for your Small Purple Widget product. The question then becomes: Does your Small Purple Widget page also rank? If not, why not? That is what you, your users, and Google all want to rank. Everyone wants the same thing so it should be simple to fix. And if they both rank, then there really isn't a problem UNLESS - your category page doesn't rank for its own keyword, at which point you should think about adjusting the short description text (if it only appears on that one page) or removing it (if it appears on multiple pages).
As always, testing is key. You can take them out and see what happens. It isn't that difficult to add them back in on most systems.
Let us know if we're not understanding your question correctly.