I wouldn't call that a "home page" as such, since it always displays exactly one note (or none eg if there are no notes in a selected notebook, or search).
Rather it's a layout with additional panels - which the new web client does not have, but certainly could.
In the old web client you could have a side panel listing all shortcuts, all notebooks and all tags - expandable / collapsible - as well as the list of notes in the selected notebook or search (including tag search). I think that should be brought back in the new web client. Would that meet your need?