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 can 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 comforting feeling of not being isolated. Also, they broaden my sources of inspiration.