Firstly, if you're worrying about ionic solids, you should consider melting points, not boiling points!
Having looked those up they show the same trend
Na
2O < Al
2O
3 < MgO
When it comes to explaining melting points of these solids which are all
mainly ionic in nature you need to consider how ions are attracted to one another (when they have opposite charge).We can use coulombs law if you know it, if not its simple to explain:
The energy of the interaction is proportional to the charges on each ion, this explains why Na
2O has the lowest melting point, it has low charge on the metal.
The ordering of Al
2O
3 and MgO can't be explained in this way. However it is important to note that these two compounds have quite different 3D structures. The attraction between ions also varies with the distance between them in the crystal structure.
Calculations of the energy of ionic lattices is more of a degree level problem (use of the Kapustinskii equation is usually involved) and can only really give approximate values.
Polarising ions like Al
3+ certainly do lead to some covalency but in the case of oxides the ionic attractions dominate.
So, overall don't worry, the two theories you've been taught both have an effect and explaining melting points is very much about the balance of several factors.
At your sort of level the main thing to know is that ionic compounds have high melting points because they have an extended 3D lattice with lots of strong electrostatic attractions between the ions, the exact ordering of these melting points is less important!