Is there an intuitive reason for why NAND gate is a universal gate?
Now I know the maths and logic to figure out that every boolean function can be expressed using only AND and OR gates


Untrue.  You also need inverters.


  every boolean function can be expressed using only combination of NANDs.


True.  Or you can do it with NOR gates...


  But, I am looking for a more intuitive (not necessarily non-mathematical, maybe philosophical) reason as to why this has to be true (if there is such a reason at all)


What both NAND and NOR have in common is that they:


Allow you to "recognize" one unique possibility of the four in a two-input logic table.
Allow you to build an inverter by feeing the same signal to both inputs


So basically you have a "test" engine and an "transform" engine; you use the "transform" engine as needed to turn the pattern you want to look for into the pattern the "test" engine does look for, and then the "transform" engine to turn the result into what you want.

And if you need to detect multiple patterns, you treat having both of them satisfied (or neither of their anti-patterns satisfied) as the pattern to be detected in another stage.

Or just find a napkin and draw all the other gates as a collection of NAND gates...