My advice goes for no, don't switch.

You always have the backup camera in your phone; do you currently even try capturing pictures with that when your big DSLR is away? If not, I suspect the real reason why you're not taking pictures now is not the size of your camera. It's probably either lacking time or dismissing opportunities to actually take some photos. For keeping memories, phone pictures will often be good enough; they sure beat having no picture at all. If you have plenty of failed phone pictures to show, then yeah, you might need something in between.

Try carrying the DSLR with only one lens, say the nifty fifty. This is much smaller than lugging the whole kit around, and not really that much bigger than a mirrorless would be, and solves quality issues in genres where a camera phone comes weakest (low light, action). If needed, a 40mm pancake lens would even give you a more compact system.

Changing camera systems is a pretty major investment compared to starting using what you already have. With a baby coming, there are already many expenses and reduction in income per capita to worry about.