I just wanted to add something to the discussion about the utility of adding additional basepoints. It turns out this is crucial for understanding certain aspects of embedding theory. See the bottom of this answer for some more commentary on this.
For a map of spaces $A \to B$, let $T(A\to B)$ be the category of spaces which factorize this map. This has objects given by factorizations $A \to X \to B$ and morphisms maps $X \to X'$ which are factorization compatible in the obvious sense. Let's consider the case of the constant map $S^0 \to \ast$. Clearly, an object of $T(S^0\to \ast)$ is just a space with a preferred pair of basepoints.
Then unreduced fiberwise suspension can be regarded as a functor $$ S: \text{Top}(\emptyset \to \ast) \to \text{Top}(S^0 \to *) . $$
Now a desuspension question in this context asks given an object $X \in \text{Top}(S^0 \to *)$, is there an object $Y \in \text{Top}(\emptyset \to \ast) $ and a weak equivalence $$ SY \simeq X ? $$ More generally, I've gotten a lot of mileage out of the fiberwise version of this question.
Given a space $B$ we can consider the unreduced fiberwise suspension of $\emptyset \to B$ as the projection map $B \times S^0 \to B$ (here unreduced fiberwise suspension of $Y\to B$ means the double mapping cylinder of the diagram $B \leftarrow Y \to B$, or concretely, it's $B \times 0 \cup Y \times [0,1]\cup B \times 1$.
Unreduced fiberwise suspension is then a functor $$ S_B: \text{Top}(\emptyset \to B) \to \text{Top}(B\times S^0 \to B) , $$ and one can consider the problem of whether an object $X \in \text{Top}(B\times S^0 \to B)$ can be written as $S_B Y$ up to weak equivalence.
Why I care about this problem
This problem naturally arises in embedding theory: if $P \to N \times [0,1]$ is an embedding, where $P$ and $N$ are closed manifolds and if $W$ is the complement of $P$ in $N \times [0,1]$ then $W$ is an object of the category $\text{Top}(N\times S^0 \to N$) and a necessary obstruction to compressing $P$ as an embedding into $N$ is given by the problem of fiberwise desuspending $W$. Furthermore, in certain instances (cf. the reference above) the existence of fiberwise unreduced desuspension suffices to finding the compression of the embedding. (This story is explained the paper: Poincaré duality embeddings and fiberwise homotopy theory Topology 38,597-620 (1999).)