"Walk of shame" is a figure of speech.
Just cause the sign says 35mph doesn't mean you need to do 35mph either. If you can't keep up just walk on the sidewalk like a pedestrian. Where's the shame in that?
You don't think I have hills to deal with where I'm from?
Try world class mountains. We like to climb these for fun.
Can we keep up with the surrounding traffic? Heck no!
But we don't let that discourage us, we just keep trucking.
Hills are a part of cycling/commuting, you either embrace them, or avoid them.

At the end of the day, riding on the sidewalk is against the law in many countries.
I hate that society look @ cyclists as villains. Encroaching on an area deemed safe to pedestrians just feeds the hate.
I don't encourage adding to the growing prejudice we as cyclist take on cause of a few bad apples.
What's wrong with walking with your bike on the sidewalk?

Cycling is made to be fun, & a form of exercise.
If we keep pushing the borders, we'll find our borders receding.
You are not doing anything wrong going @ a pace you can handle going up a hill. If drivers have complaints, let them air them to congress. Make our roads more bike friendly, improve our infrastructure to allow cyclists, vehicles, & pedestrians to live in a happy medium.