My solution: bought a DSLR when my first kid was about to be born, already had a Sony digicam at the time. Eventually upgraded to a DSLR that could do decent video too.
But you know what? I bought a Nikon waterproof/shockproof CoolPix back in 2011, and have discovered that almost ALL my video is done via that, along with some of the best family photos. For a DSLR to work well, you have to take the time to have the camera on hand, compose the frame, meter the light, etc. I still take it on hikes, and to places where I can use a tripod and get posed shots -- but for anything candid, and for almost all video, the CoolPix comes out. Because it's shock resistant, I can even hand it over to my kids who have shown that when you don't restrict how many photos they take, they can take some amazing shots (this started even at the age of 3).
This leads to the next bit: take lots of photos, but keep only the ones that tell a story or look great. At the beginning, this will be REALLY difficult to do; but eventually, you'll get so that you can come back from an event with 400 photos and an hour of video, and quickly shrink that down to 10 photos and 5 minutes of video stock (which will then get further trimmed later when you assemble it into an easy to watch montage of 2-5 second clips... right?"
Then comes storage: I have a mirrored rsyned backup of my computer where all this data ends up (2x5Tb right now), and also 2 1TB portable drives that cycle through a safe deposit box. So far, all my edited down photo and video data fits in 1TB, so this works for me.
So the TL;DR is:
1) the best and most memorable shots will come from what you have on you at the time that you aren't afraid of damaging.
2) Take your photos/video in bursts; take a bunch, keep a little.
3) Take the time to do events and parts of events with the camera stowed, so that you can actually have fun things to remember as a family (instead of a record of you recording what everyone else is doing). Use the photos and videos as triggers for events, not to document every living minute. Your memories will be much more vivid and interesting than the photos and video, even if they become less accurate with time.