I have this thing... I never walk on a road bike. If I have to stop for a moment so be it, but then I keep riding.
I love to MTB and I had to get over my thing about walking. There are just times walking makes more sense - that's why the shoes are so comfortable and the cleat recess - so you can walk! The fact that you keep moving is the key, doesn't matter if you are riding or walking, you are moving. Even the most hardcore will get off and walk if a section taxes his or her skill or endurance. 'sides think of the exercise of pushing a 30 pound bike up a steep hill. Builds up the arms and leg muscles!!!
Of course, there is no shame in trying to ride something first. The key to riding up a steep hill? Know your limits. Don't try and muscle or power your way up. Get into a low comfrotable gear and work on a steady, cadence and take the steep hill as a opportunity to rest and go easy. Once I stopped trying to power up I was amazed how much better my climbing got!