Does using a bibliography software actually save you time aside from when converting citation style?

It depends what you mean by using bibliography software. I think of bibliography software as doing three things:

1. They help you organize, search, and find your references. While Pubmed and Google Scholar are quite efficient at finding references for my field, I often prefer to search my own library of papers I am familiar with when looking for a reference. I use JabRef for this purpose and it saves me loads of time even when not converting citation styles.
2. They help you create a reference list at the end of a manuscript/grant/etc. If you have a database you simply need to tell the software what papers have been referenced and what format you want the reference list in. This saves you time when you convert styles (and the first time you create a list). I don't think it really does anything else. The key is that in my opinion ALL bibliography software does this stage well for ALL styles. I see no reason not to use bibliography software to create a reference list at the end. This is the section where it is easy to make minor mistakes and can waste a lot of time getting the style correct
3. They help you with formatting in text citations. This is where most of the software falls down. In text citation styles have a lot of variability (book, chapter, article, first time citation, subsequent citation on a page, citation in a foot note, etc) that make automation hard. Defining an automated system that can implement an in-text citation style is no small task. Even if you can create such a definition, many publishers have small in house tweaks. Create software that is fully compliant with a style and allows for tweaks to be made easily, is even harder. If you are lucky enough that your software has the style you need, or that your target publishers are easy going enough, then using bibliography software for your in-text citations is a no brainer. If you are not so luck, you may not want to use that feature.

In summary I would always use bibliography software for 1 and 2, but only for 3 if I am lucky.

How easy it is to manage references depends a lot on your working conditions.

If, for instance, you're an academic in a humanities field, where the "standard" bibliographic style is the Harvard or MLA styles, where you just quote an author's name and the page number, then bibliographies are relatively simple, since citations are straightforward, and the bibliography itself is simple and can be created on the fly.

If, on the other hand, you are working in a field such as mathematics or physics, which uses the "numbered" style, putting together the bibliography can be a royal pain in the neck. You need to add a new reference at the beginning of the document, and now all of the reference numbers have shifted throughout the rest of the paper. Then having a tool that will do the referencing for you automatically is a major help.

IF you need to use a package, and your choice thereof is up to you—you should find one that best suits your needs. But the important thing is finding a method that works both for you as well as for any colleagues you might be working with in the near future.