  The online joint replenishment problem (JRP) is a fundamental problem in the
area of online problems with delay. Over the last decade, several works have
studied generalizations of JRP with different cost functions for servicing
requests. Most prior works on JRP and its generalizations have focused on the
clairvoyant setting. Recently, Touitou [Tou23a] developed a non-clairvoyant
framework that provided an $O(\sqrt{n \log n})$ upper bound for a wide class of
generalized JRP, where $n$ is the number of request types.
  We advance the study of non-clairvoyant algorithms by providing a simpler,
modular framework that matches the competitive ratio established by Touitou for
the same class of generalized JRP. Our key insight is to leverage universal
algorithms for Set Cover to approximate arbitrary monotone subadditive
functions using a simple class of functions termed \textit{disjoint}. This
allows us to reduce the problem to several independent instances of the TCP
Acknowledgement problem, for which a simple 2-competitive non-clairvoyant
algorithm is known. The modularity of our framework is a major advantage as it
allows us to tailor the reduction to specific problems and obtain better
competitive ratios. In particular, we obtain tight $O(\sqrt{n})$-competitive
algorithms for two significant problems: Multi-Level Aggregation and Weighted
Symmetric Subadditive Joint Replenishment. We also show that, in contrast,
Touitou's algorithm is $\Omega(\sqrt{n \log n})$-competitive for both of these
problems.
