How I do it ...

When using my long zoom lens, I almost exclusively shoot birds in Shutter Priority mode and at as high a shutter speed as the camera/lens/light will allow.
As others have mentioned ... don't use a shutter speed that is lower in number than your zoom length. So 300mm means 1/300th absolute minimum. I'm much happier at 1/800 or higher if I can get it.

I've very rarely ever used a tripod for shooting birds. But I do have image stabilization.

I use a single focal point not the whole 9 points. I obviously try to get that single point on the birds eye.

I use AI Focus or AI Servo which are Canon modes for making the camera re-focus when it sees movement even after I've composed the shot.

I don't expect sharp photos unless my subject is reasonably close to me. I'd be happy to get that pelican of yours sharp at 300 - 400mm if that was as close as I could get. But at 300mm I'd want the pelican to at least fill half of the screen if I could.

Regardless of the shutter speed, the flash is very quick so I find it's helpful in stopping the movement if it's not going to ruin the shot or be effective at all.

I haven't found that a cheaper lens can't do the job. It may be a little slower on focus, and a little more soft at the longest zoom.