The Queen isn’t going to get involved over Archie
If something hideous happened to William and Kate wanted to take him off to the other side of the world to live as George Middleton then we might see it
Not a chance she’ll risk testing out if that old rule will hold in court over Archie
That said - Archie using no title is nothing to do with that. He’s still in the line of succession (he’ll stay there if Harry rules himself out, similar to how Prince Michael of Kent was out for years for marrying a catholic, but his children were still in) and he’ll still be entitled to use the title HRH Prince Archie when his grandfather becomes King.