I have a similiar issue: In my Windows 11 system (upgraded from Windows 10) the search index is paused after each Windows system restart. The "Windows Search" service (WSearch) has status "Running" but the "Indexing Options" dialog box shows "Indexing is paused". Restarting the WSearch solves the problem until to the next system restart.
Two things are annoying:
(1) My restart problem
(2) A missing "Resume" button in the "Indexing Options" dialog instead a grayed-out "Pause"
The fastest way to restart WSearch:
(1) Open Task Manager (Ctrl+Shift+Esc), be sure More details in the bottom is clicked (or Fewer details is shown there)
(2) Select Services Group, sort by Name
(3) Find "WSearch" in the Name column and select with right mouse button and select "Restart"