The kind of cooking is called "churrascaria" in Brazil, "asado" in Argentina. Same thing. The grill you saw is probably a custom churrascaria grill.
The actual cooking mixes heat energy from the glowing coals (radiant heat), from the air heated by the fire (convection), plus just a little bit from the hot metal going through the meat (contact conduction). I bring this up so you can see that there are American (or at least Californian) grills that do the same thing.
Probably your best choices are the "Santa Maria" style grills from Santa Maria BBQ Outfitters, and Peoria Custom Cookers. Klose makes something similar, but Klose is pricey. You'll need to get your wife very drunk. A few Southwest makers will custom build a "fajitas grill." Alla time same same.
The method requires a big enough fire to get a lot of radiant heat under the cooking grate, and enough space between the fire and the grate so the meat doesn't get much crust or ever get hit by a grease flare up. Which in turn means a big fire because you're cooking "open face" all the way, which means lots of space. And so on. Hot fire. Check. Lots of space. Check. That's why the Santa Maria "swing set" style grills are ideal.
FWIW, usually the only seasoning the South Americans use during the cook is salt water, and maybe a little lemon juice when it comes off the fire. In Argentina they serve it with chimichurra -- which is more of a dressing than a hot sauce. In Brazil they offer you a bunch of options and you kind of mix your own at the table.
A marginal alternative to the swingsets, are the "crank-o-vator" grills from Cajun, Barbeques Galore (Bar B Chef Texas), and a few others. The problem with these is that most of them don't really give you enough space between fire and cooking grates, so you have to use a controlled fire and tend it. Doable but PITA. Watch out for the cheap clones from Brinkmann, etc. Don't waste your money.
Hope this helps,