I realize this is a 9-year-old post at this time, but @mudworm has pinpointed my own concerns regarding the EZ-Level system more than any other post about this product here at Houzz to date. For anyone else who may be researching, I'd like to try to address some of the points that have been raised along with hoping that anyone else with more experience with the system might want to add to this later.

• Longevity - EZ Level's cabinetry leveling products have been around since about 2009, and cabinetry leveling systems are not their only products. The fact that they are still recommended by people a decade later and there aren't any disastrous reviews of longevity on the web that I can find says a lot. The company also provides a lifetime warranty against failure with a refund or replacement of its products, though that would be of little help once you have an expensive countertop installed over the cabinetry. They also have an established manufacturing plant in Florida, USA, near their office, and sometimes having a manufacturer close to home allows for a more robust quality control program. They offer several reassuring reviews on their home website from professional installers, and you can find many positive independent reviews under the EZ-Level installation videos on YouTube.
• Weight Distribution - I don't think traditional shimming completely distributes the weight of the cabinetry all across the full perimeter as the OP suggested. Not having installation experience at this time, I wouldn't know for sure, but tutorials I have read around the web talk about shimming just the four corners, same as the EZ-Level system would do. One thing you could do to help distribute weight a little more is to order more of the EZ-Level units than the company recommends. They recommend only 1 set at any place where two cabinets are joined, but you could order 2 sets at each join and do your primary leveling with one set and then just adjust the others down to a snug fit on the other set if desired. This isn't the recommended installation procedure, though. What you do need is a flooring substrate that will withstand the force without cracking or otherwise deflecting. I'm personally installing porcelain tile that is rated for over 500 pounds of force per square inch. That's the kind of resilience you want to be looking for in a floor underneath cabinetry. You might need to look into a different sort of support system for flooring that is softer or expected to deflect over time.
• Screws Moving - Yes, that's been a concern for me that since the EZ-Level system uses screws for adjustment that the screws might move over time. However, as mentioned in the first point, this company has been around a long time and still gets glowing reviews. The best thing might be to email the company directly and see if they have comments about this. Always remember that shims can also move and shift over time, too!