This example and many other illustrate that geometric arguments  cannot always be completely replaced by algebraic ones, much like the fundamental theorem of algebra  does not seem to have a  simple purely algebraic proof. (I'm out on a limb with this statement.)

It looks to me that a large part of  the fundamental  functors  of algebraic topology have a geometric origin; think homotopy, (co)homology, cobordism, $K$-theory.  I cannot  imagine formal arguments, devoid of geometric intuition leading to such concepts.  Obviously geometric arguments alone  cannot get you very far; think homotopy, (co)homology, cobordism theory, $K$-theory without long exact  or spectral sequences.

Being a mathematical "mutt" myself, I always favor impure arguments. They give me the comforting feeling of  not being isolated.  Also, they broaden  my sources of inspiration.