We are launching a news/media website called "Rewire Daily." We will offer daily articles on success, leadership, entrepreneurship, personal development, etc. Would it be better to use the WordPress platform for such a site, or should we have a custom site built from scratch?
WordPress, hands down. Development will be quicker and cheaper than rolling your own. There are plenty of tutorials to show you how to do things so you would not be 100% reliant on your dev(s).
I be wary of Cloudflare as it does not always help - you should test and see if it's a benefit based on your host and what plugins you are using.
Host-wise, most of the 'managed WordPress' services are garbage. It would be better to find a reliable and reputable host - though this can be difficult with 99% of hosting reviews being nothing more than a collection of affiliate links arranged by the size of their commissions. WPEngine is the real deal but their costs are based on page views and their $30 plan can easily be $200 for a high traffic site. MediaTemple & GoDaddy's WordPress offerings are garbage.
If you need a more customized size, Drupal can be a good option. It offers much easier customizability out of the box than WordPress. But from the sounds of your needs, WordPress is the way to go.