These Wood Pigeons (aka feathered vacuums) are doing my head in now! Help!

I’m at my wits end with the massive influx of wood pigeons coming to my garden now. They bully all the other birds and eat EVERYTHING. EVERY. THING. 
And now we are getting collared doves which are just prettier versions of the bird I can’t stand! Disappointed

I don’t want these greedy things in my garden, but they‘re going away and bringing all their friends, and now I feel like I’m turning up to some illegal pigeon rave whenever I step into the garden as they all scatter like I’ve come to bust them, but then they return once I’ve gone. They’re costing me a fortune in bird food so the little birds can even get a look-in. 

I’ve tried the “pigeons hate chilli powder so put it on your bird food” trick. Doesn’t work. 

I know you get those ground feeder cages to only let the little birds in, but the rats will come too. I need something that stops them using my bird feeding station, primarily. 

If only that certain law wasn't brought in recently that’s allowed them to have an unchecked population boom. They’re out of control now. Far too many of them. Disappointed