Heya,
You have to weigh portability and convenience with the properties you can get with your SLR & various lenses.
A friend of mine went all out, went straight to a full frame 5DII, 100L, 135L, Sigma 50 F1.4 and kept it all clean and locked away, used it some times with the family and some events, but otherwise, it slowly but surely gave way to the iphone. For random snapshots which is the hallmark of most folk with their kids, having a big contraption is a hard thing to want to lug around and manage kids and still have a good time yourself. So, like many, the big camera deal went away and the iphone one out. My friend sold his entire setup and strictly uses an iphone. It's just convenient, instant sharing, video, stills, anything, instant and at the finger tips and fits in a pocket. For someone who needs that kind of convenience, having a big dslr will probably always be an issue.
I like to have a balance. I take out my big stuff when I am going to do photography.
When I'm just wanting to have a decent camera around for the kids, family, travel, or just general snap shots, I do not use my phone, I use a mirrorless EOS-M with a 22mm F2 pancake prime. For a lot of people who want instant gratification and convenience, this is still not a good answer. Me, I want a fast lens for low light, the ability to isolate with thin depth of field, or the ability to stop down for depth of field, while being wide enough for most anything, and have enough ISO ability to take shots in the dark where cellphones fail miserably and give muddy, grained out, blurry messes, I get clear shots that are clean. Easy enough to use by anyone, no fancy buttons or needing to know stuff, I can set it to literally just track faces and hand it to someone and they can just touch the screen and it will take a photo. It's my go to for general stuff and snapshots. I shoot in RAW. I have to still process these images. But overall, this is worth it to me, because I want to be able to print some of these if I wish.
So again, you have to balance what your needs and goals are, and what you actually want, versus what you'll be willing to compromise.
That T2i and any lens is going to be able to do a lot better than an iphone will at almost everything. But, it will take you knowing how to properly expose, not shooting in automatic, and being creative and conscious of composition and pay attention to light, temperature, etc. You will then have to process them. No instant sharing. If that sounds too much of a hassle, for a good image, then maybe this kind of photography is not for you. If you're not happy with your iphone images of your children's memories, then maybe ruffle your feathers a bit and go against the "I need an easy button" grain.
It's up to you to figure out if convenience and instant gratification is more important to you, than ultimately having good images that you can appreciate later down the road, that your kids will appreciate later down the road, etc.
Again, I do a balance. I have 7 cameras and anything from 10mm to 600mm. I take out a lot of gear when I want to do photography, and I generally go out with multiple cameras. But, when I'm not going to do photography and I just want to be able to take nice images, I take a compact. For me that's the EOS-M and 22f2.
That said, maybe consider something like a Canon G16. It's inexpensive, compact, has really great quality, and if you devote a little effort to photography as a skill and art, you will be able to get way better images from this than an iphone, especially in lower light shooting RAW and controlling exposure and ISO for your needs. If I wasn't already using my EOS-M, I'd get this G16 most likely.
Very best,