This app was mentioned in 5 comments, with an average of 1.00 upvote
I too find the 3D effect to be lacking in most SBS videos using Cardboard, but 3D games and apps made specifically for Cardboard are much better. Try this, https://play.google.com/store/apps/details?id=com.fibrum.roallercoastervr&hl=en.
Chrome experiments are also quite good for that sense of depth, https://vr.chromeexperiments.com/
Roller Coaster VR
It all depends on the which and how software is used. "Cardboard" is a label for a certain type of VR experience. Is Gear VR a Cardboard? If you use it to play 30min sessions of Dreadhalls with a controller and all the nice features the Gear VR hardware and Oculus Mobile SDK provide, then no. If you don't connect the USB plug for the external sensors and just take a short ride on a roller coaster, then yes, you use it as a Cardboard version (featuring improved lenses and a mismatching FoV) for your Note 4/S6.
This is not really a question about a specific kind of viewer. "Cardboard" has more to do with low performance, bad sensors and lack of calibration than with a manufacturing process involving either dead trees or dead algae that didn't fully rot for millions of years. FULLDIVE is a Cardboard, because it cannot improve the phone part. The lenses will be better, the comfort higher, but other than that you get the same experience as you'd get from a cheap Tinydeal clone.
Is there "potential for phone based VR"? Of course, otherwise I wouldn't spent so much time on /r/GoogleCardboard. And I'm not only referring to future generations of VR optimized smartphones. For me Trinus Gyre/VR is already a use case that makes adding padding and head straps to a cheap Cardboard clone a valid option. The use case is limited due to requiring a PC, controller and users that can stomach rather high latencies, but those who can could benefit a lot from something like FULLDIVE. I wouldn't even categorize a clone modified for use with Trinus VR as "Cardboard", this would be a (primitive) HMD. Partly because the PC does most of the actual work, although an experienced developer could achieve something similarly that runs completely on a phone, still making it an HMD due to the way it is used.
But the majority of users will only use regular VR apps that run on the phone. They will have to hold the viewer in their hands and suffer from all the problems the simple construction causes. Because of that, developers will have to create short, interruptible VR experiences that work without any additional controls. The insight that Cardboard isn't really a cheap Oculus Rift has caused a lot of disappointment, but once developers accept that there are usage limits for most users, they can focus on creating VR experiences that work well within these constraints, which will improve the available software a lot. Google's bite-sized VR definition is driven by the question what will work for a large number of users, not just a few enthusiasts, and that is why they push it with their design guidelines, certification program, SDK and other measures. It is not an absolute rule, more a "best practice".