If there is a larger electronegativity difference between two atoms, shouldn't the compound be more ionic? In an explanation of the diagonal relationship, it states that "on moving across the period, the elements become progressively more covalent" yet their electronegativity increases. What is even meant by an element being covalent? I thought only compounds/molecules could be covalent.
The main force responsible for holding together covalent bonds and ionic bonds is different. Ionic bonds are held together through electrostatics. A transfer of electrons causes one atom to be positively charged and another one to be negatively charged. In other words, electrons are given away or taken away by atoms. Covalent bonds are held together due to the delocalization of electrons, i.e. electrons are shared by the atoms. For more detail look at What makes the difference between ionic and covalent bonds? (physics.se)
There is no meaning to the statement that an element is covalent. The reason is that the nature of a bond (metallic, covalent, ionic) is determined by the elements participating in the bond. Sodium may form a metallic bond with itself, but it will form an ionic bond with halogens. Halogens will form covalent bonds with themselves but will form ionic bonds with alkali metals. Furthermore, the conditions can determine the nature of a bond. For example, hydrogen is metallic under high temperature and pressure. Just a side note, compounds can be ionic. I recommend that you read this thread explaining the difference between compound, element, mixture, etc: What is the definition of of 'compound', 'mixture', 'element' and 'molecule'?
Tldr: Wikipedia is not being exact. Saying that an element is covalent is inaccurate because the terms covalent and ionic refer to bonds not elements.
The words covalent and ionic chosen in the Wikipedia article are bad. Better word choices would be non-metallic and metallic or something along these lines.
What the article wants to say is that the elements on the left half and bottom half of the periodic table are mainly metals and thus often (not always, not preferably, just often) bond metallicly to neighbouring elements and typically have a metallic element structure, while elements on the top half and right half are typically non-metals and often (see above) bond covalently to neighbouring elements/themselves. Since the elemental structure of non-metals is usually dictated by covalent bonds, one may excuse the poor word choice there, but metallic bonding is not ionic.