360 photography is the best, but it doesn't replace 2D photography. It's a supplement. I use it to hone in on details I might've missed with 2D photography. (I also have strong opinions on 2D photography - basically that one should use professional equipment instead of cell phones - but we'll leave it at that.)
Equipment
Thoughts on Insta360 series - I used it, but it's hard to resolve details. It's useful for getting an overall feel for what's going on, but it doesn't provide the resolution necessary to see things like bolts or defects in a wall. The sensor and electronics are designed mostly for video rather than still photography. I suggest spending more on better equipment. The problem is that the better equipment doesn't take video generally, but that hasn't been an issue for me.
I personally use the Xphase Scan and the Xphase Pro X2. The scan is very slow, but produces photos with unbelievable quality. The Xphase Pro is much more expensive, but it takes photos instantaneously using 25 lenses. The image quality is lower for the Pro and there are lens fogging issues when the temperature changes. It also doesn't work in the cold sometimes. So I greatly prefer using the Scan, but for big job sites, it can take a ton of time and the Pro is the way to go. I've heard good things about the Matterport Pro but never used it myself.
Read the instruction manuals! There's a specific way to use these things. It's not like an Insta360. But even if you use an Insta360, there are ways to get better shots, like using a tripod and a delay. A delay (like 3-5 seconds) will give you time to get away from the camera, and also reduce shake from pressing the button, which is a real thing that will tank the image quality.
A tripod is a must. I go with lightweight aluminum tripods. I tried carbon fiber, but those are too flimsy and they're prone to break. A trigger ballhead is useful for leveling the camera. It's not essential, but leveling it will keep your shots level and easier to look at in a structural engineering sense.
You will need external lighting for dark jobsites. I use a high-powered flashlight placed under the camera, facing upwards. I've seen people use flash, but the flashlight does the trick for me. It also reduces exposure times, speeding up the whole process.
This whole thing, just like a professional 2D camera, needs practice and training. It's essential that you review your 360 photos after each job site and figure out where you went wrong, because there is a high chance you'll screw something up. That's why 2D photography should be heavily relied upon, at least in the beginning.
Bring extra batteries. There's nothing worse than your 360 camera running out of juice in the middle of your 2nd site visit. These things use battery quickly. The extra batteries are special orders (we're not talking AA or AAA) so make sure you get the right one. I got one that was almost the right one, but not exactly, and it didn't work. Got the right one, and it works.
Software
For software, a free viewer is Panorado. I generally don't do any editing on the photos. The 360 cameras I use take HDR exposures, meaning three shots at different exposure levels. The manufacturer-supplied software stitches the HDR images together and it works fairly well. For editing, I've heard of people using Affinity Publisher and Adobe Photoshop, but I've never seen a need for it in terms of structural engineering and inspections.
Organization
I never line the photos together as a virtual tour. I know Matterport does that automatically, but there's really no need. There's a monthly fee for that (yuck) and the way I do it only has equipment costs. However, you can get lost in some construction sites. So the first thing I do is separate the photos per floor. Then I label the photo file names with "front" and "rear." I go in a straight line generally, or in a U-shape, so the ones in the middle are going to be between those points.
Other thoughts
There is a major con to all of this. When you take a 360 photo, you are basically assuming liability for everything at the job site. So I keep it a secret from everyone that we're doing it, ethics be damned. I never supply 360 photos to clients; it's for internal use only. If they want a virtual real estate tour or something, they can hire a company for that. It's up to you to create excuses for what you're doing. Everyone is curious when they see these devices.
Sometimes, I wish I could just have a point cloud instead of 360 photos. But that's too expensive at the moment, and not something I've explored yet.