I appreciate the thoughtful, in-depth response!
This is something I’ve fantasized about myself, though I haven’t put much work into it. I really like what @Mohammad did with the popup menu in this demo, though an ideal solution would be layout-agnostic. I imagine it would still have to depend on the user defining buttons and templates to be included.
You might be interested in @EricShulman’s filter generators, and perhaps particularly in his PowerSearch. Unfortunately I didn’t find this myself until after the point at which it would have been most useful to me; these days I’m generally using a wider range of filter operators than it supports, and it’s quicker just to write the filter myself. But I do really appreciate that it shows you the full filter, as you’ve constructed it, in addition to the search results it produces.
Personally, I use (a slightly modified version of) Command Palette for nearly all my search and navigation needs—though I’d hesitate to recommend it for inclusion in a starter edition as it hasn’t been updated in more than a year, and the official version can be difficult to customize. Even so, it’s game-changing to be able to pull up a search bar with a hotkey, and I find it much faster than opening the sidebar (and then probably opening Advanced Search anyway).
Ditto! I like Favorites, personally, for the ability to sort into “folders” and quickly populate a folder with a filter.
Autocomplete, Link to Tabs, Relink, and Commander are also excellent picks (and I’d also agree that Relink should be part of the core).
Eric’s Power Search may be a good starting point!
I think you’re absolutely right that filters are one of the biggest (if not the biggest) pain points for new and even experienced users, and having an interactive model for what a properly formatted filter should look like would be a huge boon. Ideally, it would include some options for working with variables or transcluded fields, too; that might clear up some of the inevitable confusion about when and how to use each type of bracket (certainly a major hurdle for me when I was getting started). But as you said, even a relatively basic builder would be educational—and perhaps worth including on the main site, as a companion to the filter documentation.
I haven’t used it myself (and the demo site may need a little Google Translate), but I wonder if @linonetwo’s Template List plugin would fill some of this need.