As has been pointed out, the expected way to treat expressions as if all variables are real is to use ComplexExpand. And while I cannot read the mind of Wolfram developers, it seems to me that Re is not intended to deal with arbitrary expressions (beyond just applying some obvious identities). It is specifically intended to give you the real value of an expression with head Complex. I think one could reasonably disagree with that, but I think my interpretation is a reasonable interpretation of the documentation.

As for wanting to have "purely real expressions", there is a very literal sense in which that is just impossible in Mathematica (without you writing your own custom functions, that is), because expressions have no type (in the "strongly typed" sense used to describe some languages in the software industry). It is largely up to the user of Mathematica to "protect" the semantic meaning of the expressions they want to use.

It seems that Re does not use the assumptions:

In[57]:= Assuming[{x \[Element] Reals}, Re[x]]

Out[57]= Re[x]

For you purpose you can use ComplexExpand:

ComplexExpand[Re[x + I y - 1/(x + I y)]]

I am surprised by the output of the following:

Assuming[{x \[Element] Reals, y \[Element] Reals},
 FullSimplify[ReIm[x + I y - 1/(x + I y)]]]

It may be a matter of LeafCount.

{x - Re[1/(x + I y)], y (1 + 1/(x^2 + y^2))}

is what I get. Not very consistent.