One of the main advantages of hosting your own blog with wordpress and a web host is you have complete control over what you can and can't do with your blog.
When hosting with wordpress for free or on medium you're restricted in what you can do with your blog.
Having a self hosted blog is also more professional and self hosted blogs with your own top level domain name will rank better in google search.
When you use wordpress free blog hosting you won't be able to use Adverts on the blog to make money and I've also never seen any free hosted blogs rank at the top in Google search or other search engines when performing a search on something.
So if you want to make a professional blog and rank in Google search and make money then you should spend some money and start your blog out correctly by purchasing web hosting and installing wordpress on that web host.
Also get your own domain which can be registered for around $10.00 per year.