A profinite group $G$ arises as an inverse limit $G=\varprojlim G_\lambda$ of finite discrete groups $G_\lambda$. There is a nice property which states that a group $G$ is profinite if and only if it is a Stone topological group.
Now say I construct a new group $\Gamma$ out of two profinite groups $G_1,G_2$ and want to ask the question: Is $\Gamma$ profinite? Typically, constructing an inverse limit system whose inverse limit is isomorphic to $\Gamma$ is a lost cause. (This is a generalizing statement: In many cases you can construct an inverse limit system, but I've kept the "construction" of $\Gamma$ very vague for this reason.) Instead, we might try to show that the group $\Gamma$ has the Stone topology.
If we choose to work in a realm which does not admit, or hasn't yet been shown to admit, the profinite if Stone implication, then we cannot lean on the Stone topology to prove an object $\Lambda$ (notably, not a group) constructed from two profinite objects $A_1,A_2$ is also profinite. A concrete example of this situation is the disjoint union of profinite quandles: Even though the disjoint union has a Stone topology, we still do not know if a Stone topological quandle is profinite, and so cannot tell immediately if the disjoint union is a profinite quandle.
Is there a canonical way of proving that "something" cannot be presented as an inverse limit of an inverse system of finite discrete "things" without referencing the Stone topological iff profinite equivalence?
Edit: I proved today that the disjoint union of profinite quandles is profinite, so this example holds in spirit, but not technically.