There are two ways to look at transforming a coordinate system. I prefer to pick an arbitrary point and ask what happens to its coordinates if we transform the coordinate system. If we do that with the transformation $\theta'_1=\theta_1-\phi_1$ and $\theta'_n=\theta_n$ for $n>1,$ an arbitrary point ends up with the same coordinates it had before except for the "latitude."
If (for example) $\phi_1=\frac\pi3,$ the transformed latitude has the range $-\frac\pi3\leq\theta_1'\leq\frac{2\pi}3$ instead of $0\leq\theta_1\leq\pi.$ In that case, look what we're doing to the following sets of coordinates:
The coordinates $(\theta_1,\theta_2,\ldots) = \left(\frac\pi6,0,\ldots\right)$ become $(\theta'_1,\theta'_2,\ldots) = \left(-\frac\pi6,0,\ldots\right).$ Does that make sense?
The coordinates $(\theta_1,\theta_2,\ldots) = \left(\pi,0,\ldots\right)$ become $(\theta'_1,\theta'_2,\ldots) = \left(\frac{2\pi}3,0,\ldots\right).$
What coordinates $(\theta_1,\theta_2,\ldots)$ must you have in order to get $(\theta'_1,\theta'_2,\ldots) = \left(\pi,0,\ldots\right)$? Algebraically, we have $\theta_1 = \theta'_1+\phi_1 = \frac{4\pi}3.$ But what point has coordinates $(\theta_1,\theta_2,\ldots) = \left(\frac{4\pi}3,0,\ldots\right)$? Wouldn't such a point usually be described by coordinates $(\theta_1,\theta_2,\ldots) = \left(\frac{2\pi}3,\pi,\ldots\right),$ that is, coordinates such that $0\leq \theta_1\leq \pi$? But then the point would get mapped to $(\theta'_1,\theta'_2,\ldots) = \left(\frac\pi2,0,\ldots\right),$ not $\left(\pi,0,\ldots\right).$
In shortAnother way to do a coordinate transformation is to transform the coordinates of every point, you're not doing anything likewhich moves points around in space, and then alter the coordinate system itself in order to return every point back to where it came from. This works fine for translations in Cartesian coordinates, and also works well for rotations in polar coordinates in $\mathbb R^2$: just subtract $\phi$ from $\theta,$ which sends every point (except the origin) clockwise, and then rotate the coordinate system counterclockwise by $\theta$ to bring everything back.
But consider spherical coordinates in $\mathbb R^3$ as an example; specifically, consider what you wanta transformation in spherical coordinates would do to the surface of the Earth if we add $\frac\pi3$ to the colatitude. Recalling that in mathematical spherical coordinates, you'rethe first angular coordinate is measured from the positive $z$ axis downward, and that we tend to assume the positive $z$ axis goes through the north pole, adding $\frac\pi3$ to this coordinate moves things $30$ degrees (about $3333$ kilometers) to the south.
Now, since Antarctica is all within less than $3333$ km from the south pole, what happens to it? Does it just messingdisappear, or do all its points go through the pole and start traveling up the other side of the Earth? Note that if it does that, the continent ends up "inside out" (the points that were originally northernmost go through the pole last and end up closer to the south pole than other points do), and moreover East Antarctica will partially overlap with South America. Or we could say that everything that goes into the south pole comes immediately back out at the north pole; this fills in the region within $30$ degrees of the north pole, which otherwise would get nothing (not even ocean), but still has Antarctica inside out and moreover puts it very close to Greenland.
It should be clear that there is no rotation of the globe that will put things back where they came from. Everything is stretched, squashed, inverted, deleted, and/or overlaid on something else.
What you can do is to add $30$ degrees west longitude to every point on the globe and then rotate it all $30$ degrees east to restore things to where they came from. More generally, in $\mathbb R^n$ you can subtract an angle $\phi$ from the last angular coordinate, the coordinate that ranges from $0$ to $2\pi,$ and this will represent a rotation of the coordinates. But you cannot do this to any other coordinate in spherical coordinates (including the radial coordinate) and expect the result to be a rotation.