I have been taking photos for many years, amateur but always complimented for my great photos. After most of my life using a great point-and-shoot camera - a Samsung VLUU WB500 (HZ10W) I bought in Korea back in 2009 - that keeps everything in focus, I figured I'd step it up a notch and get a mirrorless, a Z50.
This is new to me and while I know the basics, there is a learning curve and no doubt some user error. I have had it for 18 months now, and while I have learned tricks, this issue is yet unsolved and losing me so many memories with blurry photos. This camera has some focus and depth of field "quirks" I hope to learn more about. In addition, video mode has awesome focus, white balance, a different depth of field, etc. which I cannot replicate in my photos, no matter the settings.
I have edited my original question for anyone who shares this struggle in the future as some of the answers are rather helpful. I have also shared solutions I have found on my own that have been working well to alleviate the auto-focus issue in lower light. This question answers a lot of these issues well, though not the reasons for photos taken while set on video mode being solidly in-focus vs. on photo mode.
I most often use a Nikkor DX 16-50 3.5-6.3 lens using F-stop of 3.5. Normal quality JPEG images at maximum image size; using Fine quality doesn't improve things as they are still not fully focused.
Trying to be on the safe side, I have used the Auto mode of my Z50. But the Auto mode often takes a second or two to focus in indoor light, despite there being a good 20 lightbulbs illuminating the room. By the time it focuses, the moment is gone. While it isn't a constant problem - sometimes it does focus very well and produce great photos - I get a lot of photos that focus only 70 or 80% even when I allow it ample time to focus on the subject. I have perhaps a 5% success rate when it comes to great in-focus photos in this environment. I often hear the motor's "bzz bzz" sound while taking videos in that environment too, ruining the videos as it refocuses. My toddler is always on the move, so photos are just constantly blurry unless she freezes in time. Additionally, it often finds random things in the background to focus on and can't for the life of it keep track of the subject while taking videos or trying to take photos. Anything diamond-shaped attracts the little yellow focus box, even if it's an inch tall 20 feet away with a person in the foreground. I always have to keep pressing on the subject every few seconds as it finds other things to focus on, making my focus issues even worse.
I have noticed that in other environments, such as fluorescent lights in an arena, my camera takes some pristine images, though mostly with my longer lens. But only when the subject's back is to the camera. If they turn around and it focuses on their eye, it is almost always slightly out of focus and I find the same can be true at home too.
I'm wondering if perhaps there is too much light - some of the bulbs are the old yellow ones and some are newer LED, all pointing in different directions. On Auto mode, my camera often uses a very high ISO that whitewashes photos and videos, as high as 8,000-10,000 even with all the lights on in front and behind the camera. 1/80 10000 isn't uncommon to see on my screen. The more light it lets in, the worse the focus is. Even when there is zero movement and seemingly perfect focus, high ISOs on Auto mode can cause the subject to appear blurry.
I went through my settings to investigate solutions, but many of the settings for Auto are grayed out as the camera automatically selects the proper settings.
Solutions that have been working
I found a set of photos from months ago where for three days, my camera was stuck on ISO 3200. I had likely been using aperture-priority mode and messed around, leaving it like that. The photos were awesomely in focus in an indoor setting, even in rooms with less light than the aforementioned room. These settings - while it doesn't always stay on ISO 3200 and does its own thing sometimes - have proven rather successful. Photos no longer have such an overexposed background, focus is quick and efficient more often than not, and it is just below the point where the image would become grainy. Plus, with the higher ISO, the shutter speed is as great as 1/250 or better, better capturing any motion without blur. Photos on this setting, even without messing with ISO numbers, use half the ISO that Auto is using, so a 1/80 at 10000 ISO on Auto is almost always half that with aperture-priority, such as 1/80 4000 ISO. The lowered ISO produces far better focus without the ridiculously overexposed images Auto gives me. If it works so perfectly on this mode with near-identical shutter speeds, why is Auto mode doubling my ISO numbers and destroying the photos? Auto is still useless for me indoors, but aperture-priority works great.
Video mode clarity vs. Photo mode
On the Z50, you can flip the switch to photo mode or video mode and it's pretty common that I leave it on video and forget to switch it back. If you hit the shutter button on that mode, it will take a photo with your video settings. In my case, Fine quality and 1920x1080.
This is annoying, but... these lower-quality photos have great focus, depth of field is better, i.e., more of the image is in focus than on typical photo mode where there are focus issues, and hyper-focus on a single eyeball while the rest of the face is slightly out of focus is less of an issue. White balance is better, by far. Essentially, these mistakenly-taken photos using the video mode are what I would expect from the Auto mode I have so many problems with. But there are no video settings. Everything is automatic and nothing can be changed aside from the most basic options. There are no perfect settings I can copy from those perfect photos as the settings are invisible.
So why am I getting such great images from video mode but my photo mode is causing so many problems?
What am I missing here? Is it my lens, camera, or simply user error? Are there obvious settings I need to change? Does white balance affect focusing, especially with so many lightbulbs and with many of them different hues?