There's still a transformer (or 5) in there. Bet on it.
Reducing idle current is about "trapping" reactive power, on the primary side... more or less. Oztules has the concepts involved well documented around here in various places (look for the ferrite EI core inductor in his posts). Both LF and HF inverter topologies benefit in the idle current department from the inclusion of this inductance.
As for the "3500-7000W"... I'll have to agree, 3500W on 12V is insane... 7kW claim, even if it's only surge, is somewhere between ridiculous and unicorn.
That's near enough 600A... to pull it off for more than the handful of internal (and typically really wimpy) array of caps could sustain, the total impedance of *everything* would need to be no more than just a hair over 20 milliohms.
Scratch "Car" immediately off the list... This barely gets you out to the terminals on even the healthiest of typical cranking batteries... "Camping" as a really close second... and hope you're not the one that has to lug all that battery and copper on to a "Boat" haha
In other words, "I call bull$#!+" LOL
I unfortunately got more or less forced into running 12V (inverter was free)... I deal with currents typically between 5 and 10% of what comes with those claims (1800W HF topology PSW, in general not exceeding 600W continuous)... and even with short concise properly sized runs, the drops are "really annoying" at best, and cause their share of problems. They're much more critical at 12V than the higher nominal voltages.
3500W+? No thanks
Steve