Let me start by saying that "covalent compound" and "ionic compound" are idealizations - no compound is actually "perfectly" ionic or covalent. That's why I added the starred comment.
You wrote, "Even in crystals...but they are not bonded* together." This is even more puzzling because Silberberg states that the processes in ionic and covalent compounds "generate chemical bonds (my emphasis), the forces that hold the atoms together in a compound".
Perhaps the problem here stems from the fact that bonds in ionic and covalent molecules - while technically both connect atoms - have different properties?
Bonds in covalent compounds keep atoms together and keep them in (more or less) stable relative positions. In methane molecule there is a a central carbon atom and four hydrogen atoms arranged in such a way, the angle between H-C-H is always 109.5° and the distance between carbon and any hydrogen is around 109 pm (they do vibrate slightly, but for now let's just assume the covalent molecule is mostly rigid). When the methane freezes these molecules create a crystal, when the crystal melts, exactly the same methane molecules are recreated.
In ionic compounds there is no rigid structure, unless they are in solid, crystalline form. When they get melted ions become separated, a bit like Lego bricks. When they are crystallized back they will create structure that has exactly the same parameters, but every ion will occupy some new, random position. It is a bit like using the same Lego bricks each time to build exactly the same looking structure, but not paying attention which exact Lego bricks goes where, after all they are all identical, so it doesn't matter.
In the ideal ionic crystal the only way ions are kept together is because they are charged spheres that attract each other. This attraction is a form of bonding, but it is completely different from the way atoms are kept together in covalent molecules.