one small thing that might improve upon this method in some cases is to use focus peaking, if your camera offers it. that will help you more quickly judge if/when your subject is in focus, which can be especially helpful for objects that hover, even if only momentarily. so if a dragonfly flies over and hovers for a second, and you see the peaking indicators, take the shot(s)! if not, then find the next opportunity.
yes, you definitely have to pay some money for a camera with very fast autofocus. however fast your camera is in the autofocus department though, you can improve the chances that the camera will autofocus quickly if you use a large metering area and shoot against a plain background like a blank sky with the sun at your back.
if you can shoot 4k video, you can get screen captures that are effectively 8MP photos. 1080p video translates to only about 2MP photos (much lower quality, but perhaps enough for identification). just keep in mind that an out-of-focus video is going to translate to an out-of-focus photo. so i think the way i would implement this is to set your camera on a tripod and focus on or around a particular perch. then start the video and walk a few meters away to wait until something flies through on its way to the perch. (you could probably shoot with a second camera while you’re filming on the first one, if you have a second camera.) if you can control your camera from, say, your phone, that would allow you to start and stop the video capture from a distance (and even adjust focus and other settings) so that you can increase the chances that what ends up in the video is worth keeping.
the males do this, but often the females behave a little differently, coming over to where the males are only when they are in the mood. so occasionally you may want to wander away from the water where the fast-moving males are to find females moving around a little less frenetically.
one more thought: some cameras (especially on phones) may allow you to shoot with HDR or similar fancy tricks that may make some shots look better but may make shots with a lot of motion or fine detail look a lot worse. so know when to turn off those kinds of options.