Why not make GPE positive? I was teaching a student about negative GPE with the formula:
$$U=-\frac{GMm}{r}$$
He wasn't very happy with why it was negative and came up with this formula instead:
$$U'=-\frac{GMm}{r}+Bm$$
where $B$ is the GPE a 1km mass would have on the surface of Earth (i.e. $U=-\frac{6.67\times10^{-11}\times6\times10^{24}\times 1}{6371000} = -62.8\times 10^6J$)
It's basically shifting the red graph upwards until, at the radius of Earth, the GPE is O. This has the benefit of being very intuitive for students because once again, the ground has GPE=0, above the ground is positive GPE and below ground is negative. It's basically just like $mgh$ but with different letters.

The thing is, I really don't see what's wrong with this method and it makes all of GPE far more intuitive. I tried to rebut his arguments, but I couldn't win.

"But the radius of earth changes depending on where you are!"

At astronomical levels, it really wouldn't matter if the Earth's radius was a bit more at the equator than on the poles.

"But this formula would need to change on every planet if we colonised
other planets!"

We'd also be using other values besides 9.8 so it's probably fine.
All I could really say was that it was a bit uglier and more effort to do than the original equation, but I must admit, it is far more intuitive. Why might we NOT use this new formula for positive GPE?
 A: There's nothing wrong with this mathematically.
The intuition behind setting a boundary condition that the gravitational potential goes to zero asymptotically (rather than at the surface of the earth) is that the gravitational effect of the earth should become arbitrarily small as you go very far away from the earth. If you like, you can imagine that if the earth (and all other bodies in the universe) were not there, the gravitational potential would be zero everywhere. By adding in the earth, the potential "deforms" into a well with the earth in the center, and vanishingly small potential far away where the earth has no effect.
However since you can always add a constant to the energy there is nothing mathematically forcing you to use the above choice, it is just a useful convention. (Indeed, you could also say that it is arbitrary to say that the gravitational potential should be zero if no matter is present -- all you can really say is that the potential is a constant -- but arguably having zero gravitational potential in the absence of matter is an intuitive choice).
A: We could. But it's just an extra term to drag around. In the end, what matters are changes in GPE, so that extra $Bm$ constant term would just end up cancelling with itself.
The reason that GPE is negative is because we always need a reference point -- we must define the potential energy relative to 'something.' In this case, we chose the reference point to be when the gravitational force is zero. That happens when $r\to\infty$. Therefore, potential energy is  defined to be the energy it takes to get the mass $m$ from $R=\infty$ to $R=r$.
$$U=-W=-\int_{\infty}^R \dfrac{GMm}{r^2}\;\mathrm dr=-\dfrac{GMm}{R}$$
If you look at the $U=-\dfrac{GMm}{R}$ equation, no matter what constants you add at the end, there will always be some range of $R$ where the GPE is negative. Unlike, for instance, the tempareture scale, where we can shift the Celsius to Kelvin by adding a mere 273.15, we can't do that with the GPE, due to its asymptotic nature.
