Simplifying complex expresion (without ComplexExpand or User defined rules)

Many times, I’ve struggled with simplifying complex expressions, such as this extraordinarily simple expression, and Mathematica wont do it:

$  Assumptions = {l>0,a>0,a \[Element]Reals, l \[Element]Reals} Conjugate[(Exp[I a])^(2 l)] //FullSimplify 

Where Mathematica returns the same thing with no simplification. I told it the assumptions which make it very obvious to just change the sign of the exponent, and I don’t understand why it doesn’t work.

I looked here, but it seems like they have to define their own rules to do this kind of thing. I’m also saw here that you can do //ComplexExpand //FullSimplify for this expression, but it doesn’t seem to work in all cases. Why should it be nessecary and why doesn’t FullSimplify do it already? Also, I think ComplexExpand assumes that all variables are real, which was the case in this expression, but isn’t always the case. So how would I do it then?