The At a Glance widget shows only date and is randomly replaced by the Google search bar

I agree: At a Glance and Search each has its own settings. On a Google Pixel 6a running Android 14 long press on the At-a-Glance widget allows you to customise it (via Customise->At a Glance-> At a Glance config page), and a long press on the Google search bar brings up its search preferences.

I would ask the team configuring the standard launcher to check that the At a Glance widget is at the top of the (first page of the) home screen, and the Google search bar is at the bottom of (every) home screen page (below the fixed icons Phone, Messages, Chrome, Photos, and Camera). The Google search bar still hides the At a Glance widget at the top of the screen in build FP4.TP29.C.0101.20240121 and At a Glance (even as an added widget on another Home screen page) has never worked for me on my FP4. And on my FP4 the Google search bar has no settings to configure, at least not by a long press on another instance of the search bar on another home screen page.