Both the inverter specified on this and (of course) the airplane power plug are highly protected against overheating, overpower, and lots of other things.

Now, there's a big jump between not being on the "approved" list of AA and bringing a plane down.

How many times did you bring in carryon(s) whose size and weight were well above those on the respective "approved" list?

From an electic standpoint, inverters are just fine; some airline (I forgot which one) got wise enough to forego the whole empower/cigaret lighter mess and installed them right into the seat, providing 110v/240v to anything you might want to plug in.