That's the struggle. The best gear for family photographs is really the one you have on you and the one you'll use. As you pointed out, if you had a big bulky thing with settings to fiddle with, would you miss the moment? Probably true! If you were prepared, could have you gotten a better photograph? I don't know. Maybe one without blur, grain, etc sure, but "better" is too hard to define. A good photograph doesn't have to have the best quality in certain ways--the best photograph tells the story. I think its hard sometimes to split away from pixel peeping, worrying about grain or not, sharp or not, blurry or not, etc, and instead go back to focusing on what the photograph's purpose is, which is to basically hold a memory and re-tell the story. You see people take photographs and induce grain, purposefully blur them, basically degrade image quality, for the sake of what the photograph evokes due to that.
This exact dilemma is why I think my favorite thread on POTN is the family photojournalism thread. It grounds me and reminds me what photography is for, from the perspective of family use and purpose. I had to remind myself that you don't need thin depth of field. You don't have to have perfect exposure. You don't have to have color. It's going back to the story telling of photography, and less worrying about technicalities of image quality and attributes. Lose the gear side of requiring something specific to be able to do photography and instead work with the moment, capture the moment, and do it with what you have, not what you think would do it "better."
For example, I recently started wanting to get a monochrome camera for this. I have a monochrome sensor camera that I use for astro, for the sensitivity of true monochrome. The only way I can get that in a system like a little camera for terrestrial use is basically a Leica. Talking about costly small form factor! Gah. But the reason I was interested in it was for simplicity. Color has a psychology attached to it; I can push reds and greens and blues in a photo with saturation and contrast and most brains will have a positive response, we're wired to, but does that mean its good? No. I feel color can really distract from really studying a photograph without the brain fooling around with being attracted color. I certainly don't need the cost of a Leica just for this purpose and carry that around to get damaged. Too expensive to be a beater. So a 5Dc is my beater, and I go mono in post. Sigh!
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Some of my favorites to tell that story recently (stuck in the house with all these storms lately). They tell the story, evoke a memory, there's more to a photograph than technical stuff. Some just "work" because of what they stir up in us. Family photography is that for me. They grow up, split up, etc. I want the story & memory. I know I won't care if it's not perfectly sharp, or grainy, when I just want to remember them.
In the 2nd image, you can't see it at this angle/resolution, but the image on the fridge under the X magnet above her head is the same composition and position when she was a few months old getting into this fridge as a toddler, printed and still on the fridge to remind us of the time. She's still doing it, years later. Still capturing the story. Story within a story, granted I would be the only one to know it, but on my full res photo I can see the photo still in this photo and remember that one too. I love that kind of stuff.
IMAGE LINK: https://flic.kr/p/WXiMiu 
IMG_8422 by
Martin Wise, on Flickr
IMAGE LINK: https://flic.kr/p/WEmbRj IMG_8417 by
Martin Wise, on Flickr
... Or times when I was shooting manual focus, and those moments that remind you of things you just cannot recreate if you tried:
IMAGE LINK: https://flic.kr/p/vtZVD5 IMG_4518 by
Martin Wise, on Flickr
Very best,
