When it comes to photos of birds in flight, few other things matter if the final image is not sharp. If you would like to make sure your photos of birds in flight are nice and crisp, check out this fantastic video tutorial that will give you a range of tips as well as a nice discussion on what makes a compelling image of a bird in flight.
Coming to you from Jan Wegener, this excellent video tutorial discusses what it is that makes a good photo of a bird in flight and how to ensure yours are nicely sharp. One mistake that photographers commonly make is using too slow a shutter speed out of a desire to try to keep the ISO as low as possible. Of course, lower ISOs result in better image quality, but if your subject is blurry due to motion, it will not really matter, and it takes particularly fast shutter speeds for a lot of birds in flight. You can always perform noise reduction in post to combat the effects of using a higher ISO setting, but once a photo is blurry, there generally is not much you can do to fix it. Check out the video above for the full rundown from Wegener.
Don't short-change the Sony a7iv! It is an affordable alternative to the a1. Bird Eye autofocus works well, the 33Mp sensor is way better than my previous body (A7Riii). Using ultra-fast cards solves the buffer problem. And, with the 200-600 G OSS zoom (you did not mention the awesome internal mechanism that means no extension when zooming!), it is the perfect small budget setup (yeah: I cannot afford the a1!) (image is reduced size to fit limit!)