A used lens can be a great way to save some money, IF you are able to return the lens if it is not satisfactory, or not performing properly. So, the answer really is, it depends. Buying a sight unseen lens, is an act of faith. It can be helpful if you know the buyer, and trust them.
I agree that 200mm is way too short for birds - LIke others who have ansered, I do have bird images captured with 24-105s and some 70-200s, but MOST wild birds flight radius is such that you just can't get close enough with less than 400mm. My first bird lens 15 years ago was a 300mm f2.8 to take to Bosque, and I was frustrated no end, it was just too short even on a crop body, let alone FF body. As Shay said, a 300mm prime, pairs up with a 1.4 or 2.0X Tele-converter pretty well, but a better solution is to just pony up for the reach you need.
As you discovered a new Sony 500mm f4 is pretty breath taking when you look at the price. But you can find some used Canon 500 F4s for half that I suspect if you are patient.
Tamron makes a pretty nice 150-600mm G2 lens which I have used for thousands of frames. One of my concerns about non-OEM manufacturers lens, is how the iris diaphragms stand up to heavy high-frame rate use for extended periods, and my Tamron had endured thousands and thousands at 14 frames per second on my 1Dx Mk II. It is not as sharp as a Canon 600 f4, but focused carefully and sharpened properly in LR, it gives me pretty credible images, like this one which is about 1/3 of a frame at ISO 12800
The Sigma 150-600s get good reviews too. Canon 400 primes whether glass or DO all seem to be very good, and cost less than the Canon 200-400. One other factor to cosider about lenses is their weight; fast long primes tend to be pretty heavy. There is more than one cost to that large light gathering aperture. Heavy lenses tend to be harder to follow birds with, unless they are on a gimbal on a tripod - even more weight and lack of mobility. I use a gimbel and a tripod when I have to, but some species are very hard to track other than hand held - SE owls to not encourage tripod use, at least in my hands.
I have a couple comments about exposure, and lighting and then I'll stop. When shooting a bird on a stick, it is good to keep in mind that a bird can rotate its head to the right and left, and swing its face from sunlite to shade in an instant, and your exposure will vary by about 3 stops when this happens, which means that you have to pay close attention to the lighting if you choose to shoot in Manual Mode which many bird shooters favor. Shooting most birds in flight, unless the sun is at your back, means the sky is probably brighter than the shaded bird is - you may need 1-2 stops of positive exposure compensation.
I have been using Manual Mode with Auto ISO on my cameras for several years, and I would rec its use with your Sony. This lets me have absolute control of my shutter speed which is very critical with birds in flight, and my aperture is also fixed so that it doesn't alter my DOF, and my ISO if free to roam up and down as needed to achieve the correct exposure. Yes, higher ISOs will bring some noise, but a blurred image is fair less appealing than a sharply focused image with a tiny bit of noise, and modern bodies, Lightroom, and NoiseWare can deal with most noise these days. Manual Mode with Auto ISO was recently recommended in the latest issue of Outdoor Photography.
One other point is vantage point. Almost all images of people, children, or animals are usually best when shot near the height of their eyes. If you will look at the images folks have posted on this thread, almost all of these images were captured with the camera very near to the height of the birds eyes. That means many of them were shot from the prone position - like the eagle reflection on the water. I have an aquaintance who is a bird photographer in the UK, and that is the first thing he looks at in almost any animal image he critques - are the eyes clear, sharp, and is the viewer at the animal's eye level? This frequently means that the lens axis is just a few inches above the water level for many waterfowl shots...And height - a tall stand or column - can sometimes help you photograph birds in flight from their level or even from above them which can be interesting, We don't usually get to see the tops/backs of birds in flight from above.
Another image from the Tamron 150-600 G2 taken during daylight at ISO 1000, 1/1600th, f8