Dan, I'll try to answer all your questions. You've already had some excellent responses thus far.
You asked how many of us photograph birds. For me, most times, when out shooting wildlife, I use my FX based camera, at this time a D4, with my Nikkor 500mm f/4 lens, on a Gitzo Systematic, series 3 tripod, with RRS ballhead and Sidekick Gimbal adapter.
Periodically, I may use one of my Nikon teleconverters, but not very often. The use of teleconverters, while extending your "reach" by multiplying the lens' focal length has some real downside.
Teleconverters reduce the speed of lenses. With your 1.7X teleconverter, you lose 1.5 f/stops. When you're trying to shoot with a high shutter speed, this forces you to compensate in other ways which can have a negative impact. Teleconverters can, and often do reduce the sharpness and contrast of photos. With the increased focal length provided by the teleconverter, you magnify camera movement problems. Teleconverters can, at the least, also slow down the speed at which your camera can focus.
I don't use a remote shutter release for this type of shot. I don't think they are helpful for this kind of shooting, as I'm generally using a shutter speed of 1/1000 or higher.
You said, "I am assuming that my dissatisfaction is due to vibration like from mirror slap, tripod/head stability etc." Frankly, from a sharpness standpoint, I don't have a problem with vibration from mirror slap because my shutter speed is so high when shooting birds. I don't use VR either, as at high shutter speeds it not only doesn't do anything to help, it likely would degrade my images.
If you're using your Nikkor 28-300 VR with your Nikon TC-17EII for a focal length of 510mm, I do suspect that your tripod/head combination, having looked at their specifications, is inadequate to produce a high percentage of sharp images.
Must you have a Gimbal mount? No. I use one because it makes it easier and far quicker for me to get my camera/lens set for the bird shots due to their movement, and pan for flying birds. Do you need an $800 tripod and a $400 "heavier" head? No, but it can certainly help, and there is no doubt in my mind, that your current tripod/head isn't capable of holding your camera/510mm lens steady for a high percentage of sharp shots.
Improving your technique as suggested by Dave would certainly help. Getting your shutter speed up, as suggested by Rick will also help a great deal. And in this case, getting equipment which is more suited to wildlife photography, specifically birds would really help a great deal.
The photo I attached, of a Great Blue Heron, a large bird that's not particularly difficult to photograph was made with a Nikon D4, and Nikkor AF-S 500mm f/4G ED VR (VRII, IF, N), set at ISO=800, f/7.1, 1/3200 sec.