What was your previous camera?

I think two things you may be missing are that your new camera has a much larger sensor, which will make the depth of field shorter, plus with only the kit lens you don't have enough light coming in a lot of the time for it to focus, especially indoors.
Larger apertures make for faster focussing.

Smaller apertures make for longer depth of field. Traditionally, the camera will keep the aperture open for measurements until you actually take the shot, then rapidly close it down to your desired opening as the shot is taken. This is why a 'fast' lens is still 'fast' at smaller apertures.

The thing about a point & shoot usually is there is far less decision-making needed as the tiny sensor naturally makes for a much longer depth of field, so focussing can be pretty lax [or even non-existent, fixed focus on a tiny sensor can be sharp enough from 6ft to infinity].

I can only guess that when you set the camera to video mode, it automatically uses a smaller area of the sensor, which will make the depth of field longer, giving an apparently sharper image.

Looking at some of your other numbers - 1/60 or 1/80 is far too long to capture a moving subject. You might need to be up towards 1/1000 or more to actually freeze the action.
Motion blur should not be confused with focus blur.

Additionally, if you have a moving subject, then no matter how good the new focus-tracking algorithms are compared to earlier cameras, you will probably still have to help it out by correctly following your subject yourself. Mine has a 'sport' mode where it will track a moving object, so long as I first centre it as I half-squeeze, & then make at least a reasonable attempt to move with it. if I don't, it will get distracted by something else. It does poorly if many things in frame are moving too.

On to indoor lighting. Your eyes have a remarkable ability to see in a huge range of lighting conditions. A camera has a mere fraction of that ability. Indoors, what you consider perfectly reasonable light will be right at the bottom end of a camera's capability. Figures of 1/80 & 10000 ISO tell me it is very much struggling for light.
The only real way to increase the camera's 'eyesight' is to get a much faster lens. A lens with an aperture of f/1.4 or f/1.8 would be considered 'fast'. A 50mm f/1.4 is about the fastest you could reasonably expect, but these are not cheap. Nikon has a new 50mm f/1.8 which the world appears to love, again not cheap. Also a very fast f/1.2, at nearly 2 grand!
If you wanted to save some money & have almost the same experience, then if you have the FTZ adaptor you could try an older F-mount f/1.8 DX lens, which are really quite cheap in comparison.

BTW, on a zoom lens with a variable aperture, 3.5 - 6.3, then you only get 3.5 when you're zoomed right out. As soon as you start to zoom in, the lens will quickly move towards the 6.3 end. You can see this happening in the rear screen as you zoom. 6.3 is getting very small [& consequently slow & hard to focus] for shooting indoors.

I would recommend joining a camera club. I think you could use a guiding hand getting you over some of the elementary hurdles such as the exposure triangle, framing & subject tracking. These are far easier to learn when someone is watching you shoot than trying to gather information from a book or online. I think at the moment you are struggling with some of the basic elements & you are becoming frustrated and annoyed because there's 'too much going on, all at once'.
What you bought is not a beginner camera, so even on automatic, you need to be aware of what it's trying to do, that you may or may not want.

I found when I started I got along a lot better in Aperture priority mode, with both aperture and ISO set to manual. That meant all I had to do was watch what the shutter speed was doing. I shoot a lot of wildlife, so I need to keep my shutter speed as fast as it will go. I do this mainly by lifting the ISO in lower light to 'just below grainy' [a number you will have to learn by experiment for your particular camera].

I hope some of this rambling has been useful. There are a huge number of questions, stated & unstated, in your post & some reading between the lines required, so I've tried to at least touch on as many as I can.