I personally _hate_ this "feature" of Y! Groups. All attachments sentThe "Store on site" option for attachments is actually quite useful - when it works (or worked).
As originally designed and implemented the link provided in lieu of each attachment is a permalink which allows access regardless of whether the user has a Yahoo account or is signed in. A part of the URL is random-generated so that a link to one attachment cannot be used to infer a link to other attachments; nor can the links be used to access other parts of the group's web site without sign-in.
The key advantage is that allows members who read the group's archive or digest (rather than individual messages) to have access to the attachments. With the other option (include in emails) the attachments are sent to members on individual messages, but no other members could access them.
Groups.io appears to implement the long-sought (but never implemented) "both" option - an attachment is both forwarded in individual messages and a copy kept on site for the archive. I haven't investigated what happens for members on digest (pending tonight's digest of my test group).
Another touted advantage was a reduction in the download burden for members on slow, limited or metered email connections; those members could decide if and when to download the attachments. But with the supremacy of Webmail and mobil apps these days this seems a very niche benefit. To be a real benefit this would need to be a membership option, rather than a group option, and perhaps tied to a size threshold for attachments to be included or linked.