The best way to handle this would be to enhance Connections to make the “view” pluggable. Connections already has the code in place to determine what to show based on the “view”. A simple example is the list (card) view vs. the detail view already in Connections. At first glance, it does not seem to hard to make this change while ensuring it is backward compatible, it would just take time.
With that in place, an extension could be created, that could offer the list view to the user. I’m thinking along the lines how WordPress offers a list view and excerpt view on the Posts admin page (those odd little icons by the pagination control).
Oh, looking at the DataTables Table plug-in. It does look like a way to retrieve entry data via AJAX would be required (it needs to supports pagination, searching and ordering) … trying to render a table and then have this plugin enhance it would be really, really, slow with thousands of entries.
The actual extension would require a settings screen in the admin so the columns (fields) could be selected and arranged in the desired order. The hard part here … since entries can have n-number of addresses, phone, emails and such would be how to handle that. My first thought would be to limit those to the preferred and it a preferred was not set use the first, basically limiting it to only one of each. Or, alternatively, just down all addresses (and such) in single columns with no sorting. Actually I guess both could be done, but might be confusing to have an adresses column and separate columns for city/state/zip.
If you would be interesting in sponsoring the required enhancements, I’d be willing to do the work. Of course I’d toss in the extension. I would need to write something to test the features anyway.