My switch to mirrorless was first and foremost for medical reasons. Not kidding! I had both shoulders operated on (rotator cuff surgery) at the same time. As a pro, guide, and workshop leader, I needed to be able to carry and shoot even right after my surgeries. So, I got into Micro 4/3s. I was instantly hooked, loved the size and weight and features and the lens selection.
OK Fast forwrd through recovery, and someone gave me a Sony APS-C camera (a6000) to try. Wow! For a tiny bit more money, little bit more size and weight, I had a huge jump in sensor size, image quality, and plenty good files for any high-iso work. Then, having gotten into the Sony system, I got to try the full frame bodies, and well, soon I sold all my Nikon gear, great glass and D810.
I now shoot with Sony a7rIII, Sony a9, and soon, a Sony a7III. Baldy's right, many lenses are essentially the same in terms of weight and size (you can't change physics). The popular f/2.8 zooms (16-35, 24-70, 70-200) are all pretty much the same weight at Canon or Nikon. But there are a bunch of lenses that are much, much lighter: Compare the Zeiss Loxia 21 f/2.8 to a Canon or Nikon Z Mount 21. Compare the awesome Sony/Zeiss 55 f/1.8 to Canon or Nikon's similar. So there are some lenses designed for mirrorless that are super good and super light/small.
I love the EVF. Nothing beats having all that info right there in the viewfinder. I love how Sony keeps innovating and bringing out incremental improvements. I love the sheer number of lenses available in native Sony E-mount. Adapters are available to let folks "transition" easily, by using thier Canon (or Nikon, though it's way easier to adapt Canon glass than Nikon) lenses with the Sony system. For me, I don't use any non-native lenses anymore, since there are now (often multiple) choices for every single focal range. I think the number is now over 100.
I now carry just 4 lenses: Zeiss 21 f/2.8, Sony/Zeiss 55 f/1.8, Sony 24-105 f/4, and Sony 100-400 f/4-5.6. I also own the Sony 70-200 f/4, and will carry that when I don't need the 100-400 for wildlife.