Marla

Any camera will go blurry if you put a plain white piece of paper in front of it, cameras need some kind of contrast to achieve focus. Cell phones aren't as obvious because they have really wide angle lenses but they would go blurry as well.
For your setup my tips will work perfectly. Before putting that paper down, make sure there is something with texture and contrast like the desk or an object for the camera to focus on. Press the shutter button half way until it achieves focus, then flip the lens switch to manual focus.
You will now be able to film everything and it will all stay in focus as long as you follow my other tips. Trust me, the GH5 is way overkill for this project, you just need to learn more about how to get the most out of it and how to set it up for this type of project. It will never be a point and shoot type of camera like a cell phone.
The nice thing is, after you have it all setup, if you never move the GH5, all you will need to do is press record for future videos.

Hola Marla, whilst the GH5 is a fantastic camera it's biggest weak spot is autofocus in video mode. If continuous autofocus in video is the number 1 necessity for you then I'd look to Sony or Canon cameras (each have their weak spots as well but autofocus isn't one of them!).

If you need continuous autoffocus then I agree with @Thpriest that you would do better with either Sony or with Canon.
If you go with Canon make sure it is a model that has Dual Pixel autofocus in video. My understanding (and I could be wrong so do some research) is that some Canon cameras that claim to have dual pixel autofocus might only have it at certain resolutions. 

The GH5 does have continuous AF it just sucks. So if you're willing to put up with it, it can work ok for unpaid work. It's a bit unreliable and pulses a lot, but if you already have the camera and have no budget it still "works".

Marla - is the problem that it doesn't re-focus at all, or that it doesn't focus on what you want it to?
How close to the lens is the subject, when you bring it closer?

Before you invest in a Sony or Canon (which is recommended for better autofocus) it's worth trying a couple of things with the GH5 to see if the autofocus behaviour improves:
 
1. Update to the latest firmware as AF behaviour has been improved, slightly
2. Shoot at 60 or 50 frames per second (60p or 50p) instead of 30p or 25p
3. Experiment with AF speed and sensitivity in the AF menu
4. Make sure focus mode is set to "continuous"
You might need to consult the manual to figure out how to do some or all of these things.

Another option is try bringing in more light, setting your ISO around 3200 and shooting as stopped down as possible, also manually focus at the midway point to how close you think you will bring objects to the camera and use a wider lens. If you can bring in enough light to shoot at F8.0 while keeping the ISO from going above 3200 and shoot with a 12mm MFT lens you may be able to get your setup to work for you. Not ideal, but I would try that before getting a whole new camera and lens. Also, if you output your videos at 1080P, you can shoot in 4K and zoom in up to 2x without losing resolution. The goal is to have your entire workspace to be in focus by using the deepest Depth of Field (DOF) as possible and to move the objects only half as close to the camera as you need to while punching in up to 2x to get closer shots when needed without losing resolution.
I know that's a lot of tech talk for someone who's not a camera person so I'll try to simplify....
0 - Mount a MFT lens that goes as wide as 12mm (like the 12mm-35mm Panasonic MFT lens)
1 - Set the F stop to F8.0
2 - Set the ISO to ISO 3200
3 - Set the shutter speed to 1/30 (I know this isn't 2x the frame rate but since this isn't fast moving action cheating a bit here won't be noticeable)
4 - Set the resolution and frame rate to 4K 29.97FPS LongGOP
5 - Mount the camera to whatever overhead position it will be mounted in
6 - Add as much light as needed until the exposure meter is in the middle (properly exposed)
7 - Sit a static object in the frame and flip the lens focus switch to MF. Adjust the focus until the static object is sharp. The object should be about 50% less close to the camera as the closest point to where you plan on bringing objects to the camera
8 - Film your video
9 - When editing the video use a 1080P timeline and whenever you need a closeup punch in up to 100%.
 
That's how I would film something like this with the GH5 so that I didn't need AF.  Of course this won't work if you are trying to be cinematic or have bokeh (a blurry background) or razor thin DOF (i.e F1.4).