1) On the one hand, this is purely a procedural matter and has nothing to do with safety, airworthiness, or any other drama-in-the-skies item. There are all sorts of regular, timed inspections and re-certifications that an entire airplane and most of its components need to go through. What we’re talking about is not one of these. It’s just like having an expired tag on your license plate or driving when your driver’s license has run out.
2) On the other hand, it is quite sloppy — and as with the license-plate or driver’s license illustrations, technically it “counts.” You can’t rent a car if your driver’s license expired last week (as I once learned). Your car insurance may not cover you if the registration is past due — and airplane insurance probably wouldn’t too.
This oversight is more forgivable in one way that driving with expired tags, because the rules for renewing registration changed relatively recently. (As explained here.) But you cut less slack for a business-jet operation like this that you would some rural crop duster or a weekend amateur pilot. It could in principle happen to anyone, though as an amateur pilot, I can say that in 16+ years of airplane ownership, it hasn’t happened to me. It’s no worse, and no better, than getting in a bus or chartered limo and seeing that the license plate is out of date.
Not dangerous. Just sad.