One of the issues with the category of all topological spaces is that it lacks exponentiation, and is therefore not Cartesian closed. Suppose we take a "nice" category of spaces, like compactly generated Hausdorff spaces, so that we have a Cartesian closed category.
We can then form the category where our objects are these same spaces, but we only consider based maps, where each space has a chosen base point. Then the initial object is not longer $\emptyset$, but the one point space. Thus the initial object and terminal object are the same.
In any Cartesian closed category we have $A\times 1\cong A$ and $A\times 0\cong 0$. So, if this new category were Cartesian closed, $$ A\cong A\times 1\cong A\times 0\cong 0 $$ for all $A$. Thus this category can no longer be Cartesian closed. But, what went wrong? Do we no longer have exponentiation? Wouldn't the spaces $B^A$ just be a subspace of the previous spaces, when we didn't require only based maps?
The category of nice based space and maps is not cartesian closed with the categorical product, as your argument shows.
However, it is closed with the smash product, meaning there are natural bijections $$Hom(A\wedge B,C)\cong Hom(A,F(B,C))$$ where $F(B,C)\subset Map(B,C)$ is the space of based maps with basepoint the constant map.
In this case, we say that the category is a closed symmetric monoidal category (instead of cartesian closed) with the symmetric monoidal structure coming from the smash product.