I have some various items, using different templates (all that previously had items working fine), not showing up in either a browse preview or a faceted browse preview page block unless I also call them with an Item Showcase or Item with Metadata block (using the same resource template query). I was able to call them using a shortcode (also querying the resource template), but then it didn’t obey the line limitations I had originally set in the Theme settings. That was more annoying than a big deal since I knew how to fix it, but it’s still odd compared to other sites I’ve worked on. The biggest thing is: why are my Public items not coming up in a browse block (browse preview, browse preview-->browse all or faceted browse preview)?
Can you list all of the modules you currently have running on your site? And, may we have a link to it if it is public?
If you check the settings for your site (the site-specific ones that come up when you edit your site and go to the Settings tab in the site admin) do you have the “Restrict browse to attached items” setting under Browse turned on?
What this setting does is what you’re experiencing: you only see in browses the items you’ve explicitly put onto some page. This setting is more or less a legacy from older versions of Omeka S and for most people, it probably shouldn’t be enabled.
Yup! That did it!
I did figure out that the Faceted Browse worked after I set some columns for it to show. A facepalm moment.
The site is here: Home · Retracing the Bell Route: An Archive of Cherokee Removal · Retracing the Bell Route: An Archive of Cherokee Removal.
Jflatnes’s answer proved to be the issue. I do, however, have many many modules installed and have only just begun weeding out the ones that are extraneous. Some are “just in case Boss Man wants it”. Some of them I added specifically to compare one to the other. Others I want to pick apart and see how they work since I would like to write my own modules some day. So don’t laugh at how long my list is 
:
- Advanced Resource Template
- Advanced Search (waiting on Reclaim Hosting letting me upgrade to 4.1)
- Agile Theme Tools (although I don’t think this one is installed correctly yet)
- Annotate
- Annotate images and maps (Cartography)
- Bibliography
- Block Plus
- Blocks Disposition
- Common
- Contact Us or author and report (I have questions about the setup of this one)
- CSS Editor (my favorite!)
- Custom Ontology
- Custom Vocab
- Data Type Geometry
- Data Type RDF
- Data Visualization
- Easy Admin (very close second for favorite!)
- Extract Text (may end up not needing)
- Faceted Browse
- Fields as Tags (Not installed–need to ask “Boss Man” about it)
- File Sideload
- FreeSpace
- Generic Module
- Google Analytics (Not installed–need to ask “Boss Man”)
- Group (Not installed–need to ask “Boss Man”)
- Hide Properties
- Hierarchy
- History Log
- IIIF Presentation
- Image Annotate (waiting for Reclaim Hosting to let me update to 4.1–also want to compare to other Annotate modules to see what’s the difference?)
- Inverse Properties
- Item Carousel Block
- Item Set Calendar
- Item Set Party
- Item Sets Graph
- Item Sets Tree
- jQuery Parallax
- jQueryUI
- Local Media Ingester
- Log
- Mapping (waiting for Reclaim Hosting to let me update to 4.1)
- Metadata Browse
- Necropolis
- Next
- Numeric Data Types
- Page Blocks
- Reciprocal (Not installed, and what’s the difference between this and Inverse Properties?)
- Rights Statements
- Scripto
- Share Me Baby (Comparing to Sharing)
- Sharing (Comparing to Share Me Baby)
- Shortcode
- Sitemaps
- Table
- Universal Viewer
- URI Dereferencer
- Value Suggest
- ViewerJS
That’s a lot of modules, a bunch of which I’ve never even heard of before 
No problem there, but you might think about setting up a testing instance and do the comparison stuff there. That would make debugging on the production site much easier — often that involves a process of deactivating modules and reactivating them slowly until you find the one that does not play nicely with the others. That will happen when you use a lot of third-party modules, since developers can’t possibly test for interactions with everything. We do that with the Omeka Team modules, but we can’t cover everyone else’s.
I found all of the modules here: Omeka S Modules | From Omeka Classic to Omeka Semantic. And most of them uploaded correctly using EasyAdmin, which is handy. A few of them (although I don’t remember exactly which ones) didn’t change the name of the folder, so wouldn’t install at first, but then did. At least one of them (Agile Tools) requires some additional fiddling that I have questions about though. But I’ll make a new post about that, since it’s not related to this one.
One of “my guys” (actually, the one who has hired me for all of the sites I’ve done so far) has said that he’ll get me a domain to make my own, but he’s a super busy professor, and I’ve been able to make do with just a private page to play with as a sandbox, so I haven’t pestered him about it. But I will definitely do that soon if I’m going to start digging into the meat of themes and modules instead of little surface adjustments.
Yes, that’s the most comprehensive list, but they do have some packaging issues occasionally.