I used to, back in the 90s. I wouldn't leave the house without it. Film was expensive, so I didn't always trip the shutter. But film was expensive, so I didn't feel this huge compulsion to justify my carrying it by using it every time I took it out. Having it there meant I didn't miss opportunities when they arose.
These days the only camera I carry 99.99% of the time is my phone. For grab shots it's not bad. It won't have the dynamic range of my DSLR shooting RAW, but it's... okay... Okay-ish, maybe. Those times when I really do want to get a good solid photograph of something, I always wind up a little frustrated during processing. No cell phone can replace what an SLR or DSLR can do. I can understand why you want to carry yours.
The best way to see if you like carrying your camera is to carry your camera. Just try it out and see how it works for you. Back when I was carrying my SLR, I kept it in a flip-top bag. I don't like straps because they just get in the way when working from a tripod, so mine was just a brick in a bag. If I saw something, I'd reach in and pull it out. It worked for me. See what works for you.
As far as people in Target saying anything, it didn't really come up back then. At least no one said anything. These days I still carry around a flip-top shoulder bag, but most of the time it's loaded with sound gear. It looks like a camera bag, only... no camera. People still don't really say anything about it, but I live in a tourist-heavy area. No one really notices cameras and camera bags here. It's par for the course.
Not sure how your spouse would react to your carrying a camera around on a daily basis. My wife's never said anything, but we both do photography. She's more likely to reach over and say, "Gimme that. I got a nice shot here."