I know this is a few days old, but I wanted to weigh in with my thoughts as a new parent. Obviously nobody knows your financial situation, and you can't choose photography over diapers, but I wouldn't unload my gear just yet. If you love taking pictures, you're going to love taking pictures of your baby. In months three through six, I downloaded about 1,000 frames off all of our cameras. If I had unloaded my gear or downgraded before the baby came, it would be the single biggest regret of my parenting career.
If you've been shooting top of the line gear, you're just going to be frustrated. I shoot an older T1i and I'm saving my pennies for a 6D. I can't imagine downgrading now. As for you downgrading to a rebel, about the only argument in favor of this in my eyes would be to get a T4i with the 18-135 STM, because you'll want good video. However, dedicated video cameras are cheap and having a second camera also lets mom or grandparents get involved.
To me, this is more of a question of lens selection. I would keep your standard zoom and your fifty, for sure. The only one I question is the 70-200, because you'll never be far enough away to use it. I might unload it and pick up a 100L for portraits and quasi-macro (hands, feet, etc) or maybe a 135 (everybody says it's great) with some tubes for the close focus. The other thing I suggest is more wide-angle of some sort. I won't weigh in on selection, but once the baby can move, he's going to be crawling towards the camera, and fast. I have the 20mm (on crop about 32mm) and I'm always backing up with it. A 17-40 or even a 20-35 might get a lot of use, even on full frame, and help conserve your budget.
A point and shoot is a good idea, although if you have decent smartphones and don't keep your DSLR in a lockbox, you might not use it. I got the g1x, and the image quality is great, although it's a little sluggish. However, with a compact case, we take it all over (Wrigley field, pumpkin patch, weekend trips to Grandma's) and it is great to not lug a DSLR bag with all the rest of the baby gear. We also have Elphs but the iphones take far more pictures than those.
Most important thing is just using them. Our cameras sit out on shelves, ready to go, because you can't dig through a bag when he's having a "first". Because the iphones take P&S quality snaps, I haven't stuffed an Elph in the diaper bag permanently, but my wife usually carries hers in her purse. Read Ken Rockwell's article on why your wife wants you to buy better camera gear. It's a bit sexist and nauseating, but the principle is pretty close. I was never into portraiture, but I bought a flash, I'm trying to build a DIY ringflash, and looking at all sorts of umbrellas and stuff, too. I wouldn't bother selling your 580, in fact, I'd start looking at lighting for your DIY baby portraits. We took all sorts of home-made newborn photos, and, although they are nowhere near the quality that we got from the pro baby portrait studio, they are great, they're mine, and they are great memories to have from a time when all your baby does is sleep and poop.
If none of this helps you, then just take neuroanatomist's advice. He's pretty much always right and never full of BS, from what I've read.