DEP: integrate: improve deprecation of private modules named without underscore prefix

[j-bowhay]

Reference issue

closes #17771
towards #18279

What does this implement/fix?

Currently some of the deprecation warnings for private modules named without underscore prefix are misleading. For example:
Before this pr:

In [1]: from scipy.integrate.odepack import ODEintWarning
<ipython-input-1-0b0b6117d43a>:1: DeprecationWarning: Please use `ODEintWarning` from the `scipy.integrate` namespace, the `scipy.integrate.odepack` namespace is deprecated.
  from scipy.integrate.odepack import ODEintWarning

In [2]: from scipy.integrate import ODEintWarning
   ...:
---------------------------------------------------------------------------
ImportError                               Traceback (most recent call last)
Cell In[2], line 1
----> 1 from scipy.integrate import ODEintWarning

ImportError: cannot import name 'ODEintWarning' from 'scipy.integrate' (/home/jakeb/development/scipy/build-install/lib/python3.10/site-packages/scipy/integrate/__init__.py)

In [3]:

After this pr:

In [1]: from scipy.integrate.odepack import ODEintWarning
<ipython-input-1-0b0b6117d43a>:1: DeprecationWarning: `scipy.integrate.odepack.ODEintWarning` is deprecated along with the `scipy.integrate.odepack` namespace. `scipy.integrate.odepack.ODEintWarning` will be removed in SciPy 1.13.0, and the `scipy.integrate.odepack` namespace will be removed in SciPy 2.0.0.
  from scipy.integrate.odepack import ODEintWarning

This also reduces code duplication and strengthens testing.

Additional information

[h-vetinari]

LGTM! (modulo a nit)