Don't believe the hype.
All gravity flow water filters are designed using gravity to provide the physical force through a media. It is important that the media remove both biological and chemical contaminants from the water as gravity forces it along. This said, you need to (1) start with a media that has the appropriate physical ("porosity") and chemical ("cation exchange capacity") qualities to accomplish this feat, (2) determine the time it will take to accomplish this feat as the media and water interact with each other, and (3) design a vessel that provides the media and water enough time (the "residence time") to accomplish this feat.
Start with what's in your water that you like out of your water. Find a media (porosity and cation exchange capacity) and vessel (that provides proper residence time), and you're on your way to wonderful water.
Getting to Berky, I once in the relatively current past, tried to pry info out of them related to their media and proper residence time design. They were not forthcoming. I do know that they rely on activated carbon (I'll discuss this briefly, below) and a proprietary ion exchange resin of some sort. This said, I'm very speculative. I need to know what is in my filter and how the design works. I hope they become more transparent in the future.
Now, what about activated carbon. I could write a book on this topic, but what I'd like state here is that all activated carbon is NOT the same. It is critically important to understand where the carbon came from -- was it municipal or medical waste, or was it "organic" coconut shells? You can find this out by REQUIRING the "material safety data sheet" (MSDS) or the "material sheet" (MS). Additionally, regular carbon or char CANNOT clean water like activated carbon. How do you know you have activated carbon? You have to trust the source and their processes. For example, I purchase my activated carbon from 3M, and it comes with a MSDS, a confirmation that it is activated carbon, and their source of carbon. It'll cost you more, but you'll protect your health.
How do I know this stuff? I'm a licensed environmental engineer, with a specific discipline in water resources. I work as a public servant for a environmental protecting federal agency and lead many humanitarian water supply projects in developing nations. This is my life. I figured I'd share with you.
What's my water situation? I live on municipal supply. I gravity drip that municipal supply through my 3M activated carbon with ample residence time, and I renew my activated carbon once per quarter.