It is unclear to me how you got from your input data to your expected output. If you are "trying to count() overlapping entries" then it would seem to me that you would only have only one value for the count. Either a range overlaps or it does not.
GROUP BY will condense into a single row all selected rows that share the same values for the grouped expressions
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Which means the only valid comparison for GROUP BY is equals. The processes of finding a single value upon which such an equality comparison can be performed is the subject of the wiki page I linked.
The only other potential query approach that comes to mind is some kind of recursive CTE.
A more structural potential approach would involve triggers and maintaining some form of master range table that evolves as DML is executed against the base table.
