If I have a frame with two wheels but could only push it by foot over say 10 random miles would I gain any kind of mechanical or other advantage.
Poorly worded but my brain hurts.
That’s the equivalent of a scooter or a skateboard. Depending on the terrain you would gain a lot from the ability to coast.
If you were going to end up at the same elevation wouldn’t the ups = the downs?
Walking you’d need effort for both up and downhills. Riding you can coast down. I can’t see how riding would increase the effort of going up enough to balance it. Unless the bike weighs as much as you do.
that would be a hobby-horse. invented because it was probably better than walking.
Pushing and coasting is more efficient than walking. Wheels are good. The same effort that gets you 3’ walking can get you 20’ on a scooter or hobby horse. That more than makes up for a slight extra effort going up hill.
If your downs are before your ups (i.e., there’s a valley between your starting and ending points), you might not even need to expend much energy on the uphill parts, either: Coast downwards and build up speed, then lose that speed as you’re going back up. The only energy you’d need to add yourself would be enough to replace whatever’s lost to friction.
Depends what you call friction, you also get rolling resistance losses from the compression or the tyre material.
or a balance bike.
They’re very popular here for the pre-school set. My son has one. It’s clearly vastly superior to walking in terms of effort output - he can go about 4 or 5 times further on the bike than on his pins. Maybe more.
I wish I’d had one of those to use going to school.
Oh wait. It was uphill both ways, so it wouldn’t have helped. 
You gain on the deal because your legs only have to push you along - they don’t have to hold you up as well. That’s basically what’s going on with a bicycle anyway. The pedals and chain are just there to let this happen at a higher ground speed. Without, you’re like a car stuck in low gear; you can do the work needed to go faster very easily, but have to work inconveniently fast to do so.
Random miles of what? If it’s just ordinary, unimproved terrain, then no, you probably wouldn’t get any advantage over walking. But then, a bicycle with pedals wouldn’t get you any advantage either. Both types of vehicles need a relatively level, smooth, hard surface.