Why can't the 2 oxygen atoms donate a lone pair and hence become binding sites? Is it because of the angle made by the oxygen and Nitrogen atoms?
You are right. I took this two pictures from Wikipedia to compare EDTA (a tetradentate) to dmg (a bidentate).
For EDTA you got this:
For dmg, on the other hand, you got this structure:
You can notice that for both of the cases, the metal complex is connected to the ligands through five-membered rings. This observation is general in chemistry: to have stable rings, you need five (sometimes six) members for those rings.
Now it is clear to see a general rule of thumb for multidentate ligands: the two donating atoms must be connected through 2 (sometimes 3) atoms. For dmg, you can see that the two nitrogens satisfy this rule, while the oxygen-nitrogen pairs do not.
For EDTA, you can see that going through the molecule, you can find an oxygen, a nitrogen, a second nitrogen and a second oxygen, all bridged by 2 atoms to each other (while the two oxygens of the carboxyl group are not connected by 2 bridge atoms, otherwise you might even have a hexadentate ligand).