Missing cursor on windows 10?

A family member has a laptop with Windows 10, and the cursor has completely vanished. It's present at the login screen, and gone immediately after. They usually use a wired USB mouse, but the trackpad is doing nothing as well. The laptop does not have a touch screen, so we have no way of interacting with things normally.

Most solutions online don't exactly match this issue, or suggest fixes that require the use of a cursor, so we're a bit stuck. Hoping someone can suggest a keyboard-only fix that we haven't tried.

EDIT Additional troubleshooting

What we've tried so far, after extensive googling and keyboard-only navigation as suggested below:

Turning it off and on again (a few times)
Pressing all function keys to re-enable the cursor (fn+F1 ... +F12)
Disconnecting the mouse and reconnecting it
Testing the mouse on a different computer (it works fine)
Connecting a different mouse
Connecting either mouse (old or new) to a different USB port
Checking for a physical "disable cursor" switch near the trackpad or on the side (there is none)
Alt-tab in case the cursor is captured by a background application
Opening task manager (ctrl-shift-esc) to try to free the cursor
Flailing mouse and trackpad wildly in frustration
Windows is up-to-date, no updates available
Device manager recognized both mouse and trackpad, and says they're working properly
Alt-tab and task manager don't show any open apps after logging in - nothing to break the cursor out of.
Opened the old mouse control panel and enabled the "locate my cursor when I press ctrl". A circle pings at the center of the screen, and does not change after mouse interaction.
Rebooting in safe mode made no difference - cursor still missing, ctrl pings at the center of the screen.
Checked the cursor to make sure nobody had swapped out the graphic for an invisible one - changing to a few different system ones made no differe. Ctrl still pings at center of screen.
No unknown scripts or apps in the startup registry

What the issue isn't:

Cursur is laggy or frozen. It is not visible anywhere on the screen.
Cursor only disappears in some apps, or while typing. It is gone completely at all times.
Deprecated / end-of-life hardware. Laptop was purchased new early 2021, less than 4 months ago.
Laptop is frozen. Laptop still responds to keyboard input.
Cursor is hidden in a corner. We checked, a lot.
Cursor is caught on a different display. No other displays connected.

Solution 1:

It's present at the login screen, and gone immediately after.

This sounds like it's caused by a user-level preference.

Check within 'main.cpl' (described below) whether the cursor theme hasn't been changed to a completely blank one as a prank.
Check Task Manager's "Startup" tab for any suspicious apps. Try to disable them all and reboot. (If that helped, then re-enable one by one until you find the cause.)

Most solutions online don't exactly match this issue, or suggest fixes that require the use of a cursor, so we're a bit stuck. Hoping someone can suggest a keyboard-only fix that we haven't tried.

What we're not able to try, because it requires cursor interaction:

Checking for windows updates

Checking the device manager to see if the mouse and trackpad are recognized

I do want to point out that nearly all things in Windows can be interacted with using entirely just the keyboard, without needing mouse or touch-screen input at all. You mostly use Tab and arrow keys to do it.

Device Manager: Press ⊞WinR, input devmgmt.msc.

Alternatively, as of Windows 10, there is the ⊞WinX menu (aka "right-click Start" menu) which offers direct access to M Device Manager.

It's also findable by opening ⊞Win Start and typing "device manager" into the Start menu.
- When the Device Manager opens, use Tab to focus the device tree, then arrow keys to navigate through the tree. Either the → Right arrow or the Numpad + will expand a collapsed subtree.
- Pressing Enter will "double-click" an item, in this case opening its Properties window. If necessary you can use either ≣Menu or ShiftF10 to access the "right-click" menu of an item.
- Within the Properties window, you can CtrlTab or CtrlPgDn to switch tabs. Then use regular Tab to navigate to a particular button, Space to "click" it.
- You can also use Altletter to directly activate whatever button has an underlined letter in its text. The underlines may be hidden at first, but will show up when holding Alt. This also goes for the menu at the top of the Device Manager window.
- For example, in the main window, use AltV then V (or just Alt followed by arrow keys) to activate the "Devices by connection" view, which can sometimes be more informative, e.g. if it happens to show everything underneath a particular USB controller as unrecognized.
Old "Mouse" control panel: Press ⊞WinR, input main.cpl.

Navigable using Tab, Ctrl+Tab, etc. as described above.
- I would in particular suggest going to the "Pointer" tab and trying to change the cursor appearance scheme. Try selecting "(None)" for the old-style unthemed cursor (which should always work), or "Windows Default" for the modern-style one.
  
  After you've tabbed to the schemes drop-down list, Up/Down arrows will work, but Alt↓Down will also expand the list. Hit AltA to 'Apply' changes.
- I'd also suggest going to the "Options" tab and enabling "Show location of pointer..." – either by tabbing around as described earlier, or by hitting AltS to activate the checkbox then AltA to 'Apply' if you are on English Windows.
  
  With this option active, hit just Ctrl alone to make Windows show a circle animation around where the pointer is supposed to be.
Command Prompt: ⊞WinR, cmd (or just find it through Search).
- If you need to run 'cmd' with Administrator rights – hold CtrlShift while pressing Enter. (Works both in Run and in search results.)
Windows updates: Press ⊞WinI to open Settings. You can also directly open any Settings section through Windows Search.
- The Windows 10 settings app is UWP, so almost everything is keyboardable using exclusively just Tab and arrow-keys. (No Alt+letter hotkeys though.) The "Back" button is AltLeft.
Switching apps:
- Press AltTab to switch between windows, AltF4 to close the current window.
- Use AltSpace to open the window control menu (for move/resize). As of Windows Vista you can also use ⊞Win↑Up and ⊞Win↓Down to (un)maximize.
Websites:
- If you want to select and copy text from a website (such as a long command), press F7 to enable "Caret mode".
MouseKeys: In case you still need to click on something that really cannot be Tab'd to.
- Press AltShiftNumLock to activate the "MouseKeys" feature.
  
  (You can later disable it through Settings → Ease of Access.)
- Now whenever Num Lock is enabled, the numpad digit keys will move the cursor around (slowly but with acceleration), NumPad / or NumPad - (minus) will choose the left/right mouse button respectively, NumPad 5 will click.
This can be really annoying to use on smaller laptops which do not have a dedicated numpad, but it still kind of works.