Should I use a not-self-hosted website for both blogging and collecting academic materials?
In academia, an amateurish-looking website can almost be worse than having no website at all. Now "amateurish" here does not mean "made yourself"; it instead refers to sites that use bad design practices (lots of garish colors, ill-chosen animations and transitions, poor design choices that make it hard to find information). However, there is not necessarily a strong link between poor design and web host. So, you can and should decide if the web host makes sense for you.

I know a number of researchers who have their websites that are hosted by sites such as Google Sites, but they do look professional enough to pass muster. On the other hand, the main issues for a web site in academia is that you are able to find the information you need. A bare-bones text-based website (which many people still use) is perfectly acceptable if you can access a researcher's CV, lists of interests and publications, and get contact information.