Do all of your custom widgets have an .svg format file named 'icon.svg' in the root folder of that widget (not contained in a sub-folder)?
That's what I'd look for first a svg file in the root folder of the widget. But I have noticed sometimes you'll get the circular don't sign like it couldn't load, and usually its because something failed to build when testing for me. Or, sometimes its because you forgot to start both server and client. Check for errors in the project/console and see if something is preventing it.
Now if you aren't seeing a draggable block for your widget, then its probably because you forgot the manifest.json
These aren't custom widgets, they are Esri out-of-the-box widgets. I have one custom widget but it has all the parts and is not related to these widgets. I have deployed it several times on my web server, but I get the same result, it used to work sometimes but now it doesn't work at all. Here is the link to the app for help https://gis.colorado.gov/viewer/?page=Map-View
Inspecting your project, I see that the icons and some fonts are attempting to load from localhost. That strongly suggests some sort of build issue.
Even though these are not custom widgets, your problem may still be in your custom widget. Errors in a .json file can affect the build in strange and unpredictable ways. I was especially look for trailing commas. I had one trailing comma that broke not only the project I was working on, but also prevented me from creating a new one. Another common error that causes build problems are folders containing no files.