I think that you're not asking the right question.
You make it sound like building a site using WordPress is something like a standard box you would get in the supermarket, whereas a custom site would be something different completely.
WordPress is a CMS (Content Management System) and websites built with this software are dynamic.
Now you can implement WordPress in many different ways. You can buy a readymade theme and use that for your site or you can "have a custom site built from scratch".
Last year I released a WordPress starter theme that can be used to build a custom site on. It basically stands in the middle of starting completely from scratch and a completely ready made theme as it already gives you a solid back- and front end framework as a basis.
If you were going to make a site without WordPress and without any other CMS, then you would most likely be looking at hand-coded HTML?
The clear advantage of using a CMS (any CMS really, including WordPress) is that it is easy to change or to add on to the content later on. There are many more advantages of using a CMS over a hand coded method, but I think that is outside of the scope of your question.
If you want to talk more about the best solution for your project, I'd be happy to talk with you.