To your first question: Yes. Cameras can just stop working. As with many things we craft, materials will age, oils for lubrication can harden, things wear and break.
However, if the drop of the camera did indeed damage more than is apparent, is difficult to say without direct comparison to a working model or without inspection of the insides.
Clarification
Let's first focus on your main issue:
Manual (exposure) mode or Auto (exposure) mode does not equal to manual or auto focus. These are separate settings.
However, depending on the camera, auto exposure mode usually controls which auto focus points are used, while on manual exposure the camera usually allows you to choose.
Test if the camera still is ok
A good test would be to try to take a picture of a text sheet and see if the camera is able to take a sharp image of that. This is just a test to see if the lens mount is still ok and the camera is fundamentally able to focus.
If there is a slight mis-focus, and that is the same in all images, check your manual if your camera can correct for back-focus or front-focus. However usually that is NOT a problem, when you are filming.
If that fails, there is something broken in the camera and you can stop testing and head for a repair.
Focus in Video mode
If that works, you need to understand, that video focus and image focus is somewhat different on DSLRs. When taking videos, the mirror flips up and the camera is using the image itself to determine if the image is sharp. This can lead to the effect that the camera is "hunting" for focus.
You have been told, that good lighting is helping here - and that is true. The only way the camera can tell when it has focus is when it sees contrasts: It will have a hard time to focus a completely uniform background vs it will quickly focus a checkerboard pattern.
So having contrast in your image helps the camera.
However, the camera is at a disadvantage as its main AF is only available when the mirror is down. I don't know how bad the situation is, but a consumer DSLR of that age will pale in comparison with any current mobile phone autofocus. Just to manage expectation.
Manual Focus
If good lighting still leads to constant refocussing, drop autofocus completely: You can also try to close the aperture (bigger f-values) to have a wider depth of field which is acceptably sharp and the set the focussing to manual. This means that you now have a limited field in front of the camera that is sharp-ish. You can mark this with tape on the floor and then making sure, your scene happens in that area.
If you use manual focus and preset the focus before filming, you can also utilize depth-of-field calculators like here https://www.photopills.com/calculators/dof
Here you can set camera model, lens (for focus length), focussing distance and the aperture to see how deep the area of focus is.
For example when using a 50mm lens and these settings:
You get this result:
So you have a corridor of around 3.5 meters at 6m distance where you are having a sharp-ish image. This can be used to rely less on the autofocus. Also note, that wider lenses (smaller focal length) have wider depth of field. So it is easier getting a sharp image with a wide angle lens.
