This is the topic that made me investigate kitchen ventilation and spend considerable time on this forum for the past 14 years. This claim is not completely specious, but has to be understood in the context of blower performance and system pressure loss.

The short answer is that if one has two hoods to compare without any ducting and without any make-up air pressure losses, one conventional with baffle filters roughly modeled on commercial hood principles and one Magic Lung with squirrel-cage blower(s) extracting grease, then the pressure loss of the conventional hood baffles may take a (say) 900 CFM rated blower down to 600 CFM of actual flow rate, whereas the VaH blowers are rated for their hoods and can be called 600 CFM blowers. The latter only see minor sheet metal transition pressure losses.

However, one doesn't have hoods hanging from the local oak tree, but embedded in the architecture, with a duct to the outside, and some means of getting air back into the kitchen from the outside -- make-up air (MUA). The baffles, when present, the ducting of exhaust, the ducting and filtering of the make-up air all have a pressure drop as a function of flow rate. Fighting this is the blower's performance as a function of the pressure across it, which is the pressure loss going from cooktop to outside and back to the cooktop.

In the figure below, one may see that all blowers have a fan curve, and this relates flow rate to pressure loss. The brown curve is the pressure loss vs. flow rate, the black curve the fan/blower performance cure. The intersection provides the actual flow rate.

This relationship is true of both Magic Lungs and conventional blower systems. What isn't shown here are the different shapes of the fan curves of different blower configurations. Data released by VaH reveals that the squirrel-cage blowers seem to be more sensitive to pressure loss, so their fan curve will be more horizontal that that above. Due to this, any significant pressure loss, such as a long duct run with bends, inadequate MUA, or both, will lead to a lower net flow rate than the higher rated conventional blower with its steeper fan curve.

For example, the Broan 332H fan curve below can handle a lot of pressure loss at 600 CFM.

So, as usual, the entire ventilation system has to be evaluated against the capabilities of its candidate system components.

It is also worth noting that VaH now also sells conventional hood configurations.

So if you have a hood that can exhaust immediately out the wall behind, and a large enough outside-to-inside MUA path (with filter and heater) the the Magic Lung may be adequate for you. Otherwise, I would argue for conventional configurations, and if the duct is long enough and passes through an attic or chase, then use an external blower with a silencer in between for noise minimization.

Searching here on 'VaH' and/or 'Magic Lung' will reveal some comments related to ease or not of cleaning; I don't have one so my doggie is not in that fight.

I have a VAH, older model, squirrel cage blowers. Cleaning it is a big PITA, done by hand since the paint reportedly flakes off in a dishwasher. Can never get the actual cages clean. In contrast, by conventional baffles in my ModernAire hood liner can just be thrown into dishwasher.

Other longstanding VAH gripe is that it is extremely noisy for the level of performance provided. You can do much better as kas points out.