Yes, and no. It's easier for me to climb a hill at a reasonable pace than it is to go fast on flat ground. I can put out more watts at the same level of discomfort on a climb than on the flat. A good climb is never as easy as a flat recovery ride, but it can be pretty easy.

One start for being a better climber is to never give up and walk. You can ride stuff that's way harder than you think, but only if you actually do it. You'll build some strength and more important, gain the technique, for standing and muscling up a climb. But only if you do it.

I have to practice the same sort of thing when it comes to flat ground and headwinds... being slow there is largely in my head.

For the pedal thing, practice unclipping on a hill when you're fresh, so you have the motion wired and can do it when you're tired. You should never stop on a hill of course, but it's important to be able to stop in an emergency. And practice starting up again. So if you do run into a hill you can't conquer, you can stop to let your heart rate go down, then get going again. No walking!