In Huber's theory of adic spaces, as utilized in Scholze's theory of perfectoid spaces, we consider the situation of a Huber (or f-adic) ring $A$, which is a topological ring with an open subring $A_0$ that is $I$-adic for a finitely generated ideal $I\subset A_0$, and a Huber (or affinoid) pair $(A,A^+)$ where $A$ is Huber and $A^+$ is an integrally closed open subring. We then take the space

$$\mathrm{Spa}(A,A^+)=\{\text{continuous valuations $x$ on $A$ such that $|f(x)|\le1$ for all $f\in A^+$}\}.$$

My question is, why do we want this particular definition of a Huber ring? Or in other words, what is lost (or gained but not wanted) if we allow $A$ to be any topological ring and $A^+$ an integrally closed open subring? I understand that a "motivating example" is the ring $k\langle T_1,\dots,T_n\rangle$ from classical rigid geometry, which has these properties, but is there anything else in particular?