With any lens of greater than 300mm focal length on a full frame camera you're probably not going to get results you're happy with shooting handheld. On your 1.6X APS-C camera, the same angles of view are provided by any lens 188mm or longer.
It is true that lenses such as the Sigma and Tamron 150-600mm telephoto zooms are weakest at their longest focal lengths. But they are still pretty good. It is also true that the 300mm f/2.8 series of lenses from Canon are almost breathtakingly sharp.
But without using proper shooting technique, including adequate stabilization, you won't be able to get results that match the capability of those lenses when properly used. You probably won't be able to tell much of a difference at all when hand holding unless your shutter times are close to your camera's fastest shutter speed of 1/4000 second, and even then you'll probably still have detectable blur due to camera movement when pixel peeping images taken with a 300mm prime.
No matter what lens and camera body one is using, 600mm focal length requires excellent shooting technique. Rare is even the seasoned pro that can shoot at 600mm handheld and not get some blur from camera movement. Yes, optical stabilization helps in this regard. But OS/VR/IS/etc. isn't perfect or foolproof at such long focal lengths.
Even on a FF camera, at 600mm much more care must be taken to prevent camera movement from affecting the photographs taken. This is magnified even further when such a lens is used on an APS-C crop body. The narrower angles of view mean that the same amount of camera movement creates more blur than it would with a shorter focal length lens that gives a wider AoV. The narrower AoVs also mean the same amount of subject motion will cross a wider area of the frame (if the subject is at the same distance from the camera). Shorter shutter times can help, but support from a tripod or monopod is almost always required for best results at focal lengths in the 300mm+ 'super telephoto' range. If one is tracking wildlife in motion at very long focal lengths, a gimbal mount that can support the weight of such lenses can be rather expensive, but is also almost indispensable.
Most of the seasoned sports photographers I know (some of whom have been published in major publications such as SI, ESPN The Magazine, the NY Times, etc.) rarely if ever shoot above 200-300mm handheld. You can get good results handheld past 300mm with a FF sensor, but rare is the photographer that can get great results at those focal lengths without more stable support. Monopods are very popular with sports shooters for 300mm+ lenses (on full frame bodies). For notable Wildlife photographers, tripods with gimbal mounts are near universal for anything approaching 600mm.
I added a photo and additional info. I do have a mono/tripod but am unhappy w/it and will be exchanging.
Hummer was approx 35 feet away (maybe a little farther), 15 feet up in tree. Using Program AE, 1/500, Ap 6.3, ISO 6400, at 600mm, handheld (although very shaky!). I only cropped and resized it.
Your results are surprisingly good considered you shot at ISO 6400, 600mm, and f/6.3 (wide open) while hand holding the camera. Just stopping down to f/8 will improve the center and mid-frame resolution performance of that lens. Of course, when you're already at ISO 6400 and 1/500, there's not a lot of room to stop down even two-thirds of one stop.
Here's a comparison between the Sigma at 600mm and the Canon EF 300mm f/2.8L IS II + EF 2X III when both are used on a Canon 7D Mark II (which has a similar sensor to your Rebel T7i/800D) at f/8. The test images were taken from a tripod under laboratory conditions. The 300mm prime is a little better, but nowhere near the price difference between the two setups.
If one compares the Sigma 150-600mm f/5-6.3 C to the much cheaper and older design EF 300mm f/4 + EF 3X III, the Sigma is clearly better. Here's the link to that comparison at The-Digital-Picture. The EF 300mm f/4 is also a non-stabilised lens, so proper stabilization using a tripod or monopod is even more critical in this case.
I'm unsure how hard it would be to move one around while trying to quickly focus on an animal/bird.
The way the best wildlife photographers get their keepers is to accurately predict in advance when and where their quarry will be. That, and a lot of patience. Set up your rig and then wait for the shot to come to you. You may not get it on your first try, or even your first several tries. But eventually you will be in the right place at the right time pointed in the right direction. Increasing the odds of being in the right place and time without spooking your subjects is what is known as field craft. It's a big part of shooting animals and wildlife.
In your case where you are mostly restricted to shooting your back yard from your window or deck, placing feeders in the right places can increase your chances of catching your desired subjects from your more or less fixed shooting positions.
You might also consider a permanent mount for your lens and camera built into a rail on your deck. It might be something as simple as bolting a good tripod head directly to the rail. If the rails are solid they might be more stable than even a very heavy and expensive tripod would be.
The next step would be to get your hands off the camera. I've got a couple of friends who have cameras they set up pointed at their feeders and trigger the camera remotely from inside a window. They either manually prefocus the lens on the feeder or use AF with a specific AF point selected that is pointed at the right spot.