That sounds like the aggregation of read counts for all CpG over the region lifts the statistical power to make a significant call. That is the problem with low-coverage data. I have only little experience with sparse datasets, but I think there is not much you can do if the data do not have the power to make single CpG calls.

There's no reason to expect DMRs to overlap DMCs. As ATpoint aptly noted, DMR calling works by aggregating information across CpGs. Typically an individual CpG will not be sufficiently differentially methylated itself, but a region will be. DMCs themselves are less likely to have biological effects unless they clump into DMRs (there are exceptions to this when a transcription factor preferentially binds to a motif with a particular methylation state).