What's the purpose of defining the inner product? Is the inner product simply a definition that conveniently helps to give the notion of the norm and distance between points or vectors in a real vector space? I understand it gives the Euclidean space a sort of 'structure', but it seems like a quite artificially constructed definition. It seems to make calculations easy, but my question is, what would be the purpose of the inner product in and of itself?

As you know many concepts such as projection of a vector on another vector, orthogonality, finding the angle between two vectors, and as you mentioned the concepts of norm and distance are easily explained by dot product.
In Numerical Analysis,  most quaratures are simply a dot product of two vectors, such as $$w_1 f(x_1) + w_2 f(x_2) +...+ w_n f(x_n)$$
You may downgrade the dot product because it is easy to compute but is indeed a virtue of dot product not a failure. 
Compare it with the cross product and you will like it.   
