Unexpected behavior of tree splits: missing values handling is buggy?

### Describe the bug

When adding a sanity check in the best split function (`_splitter.pxy`), I get a bunch of tests failing. This probably reveal a bug in missing values handling.

### Steps/Code to Reproduce

Add those lines after [this line](https://github.com/scikit-learn/scikit-learn/blob/main/sklearn/tree/_splitter.pyx#L517).
```Python
current_proxy_improvement = criterion.proxy_impurity_improvement()
if current_proxy_improvement < best_proxy_improvement - 1:
    raise ValueError(f"Unconsistent improvement {current_proxy_improvement} < {best_proxy_improvement}" )
```

And then run the following tests: `pytest sklearn/ensemble/tests/test_forest.py sklearn/tree/tests/`

### Expected Results

No error is thrown, proving the final partitionning of samples is optimal and the children impurities are the correct/optimal ones.

### Actual Results

Many tests fail. I will split them into two categories, based on the alleged cause:


1. **MAE criterion**
Errors that have likely the same cause to than those two issues: #32099 #10725. The current implementation of the MAE criterion is slightly buggy. My PR https://github.com/scikit-learn/scikit-learn/pull/32100 will fix it.

```
FAILED sklearn/ensemble/tests/test_forest.py::test_importances[ExtraTreesRegressor-absolute_error-float64] - ValueError: Unconsistent improvement -9.0 < -4.0
FAILED sklearn/ensemble/tests/test_forest.py::test_importances[ExtraTreesRegressor-absolute_error-float32] - ValueError: Unconsistent improvement -9.0 < -4.0
```

On the branch of my PR those tests don't fail.
 
2. **Missing values**
Many tests related to missing values are failing. As explained in my PR https://github.com/scikit-learn/scikit-learn/pull/32119, the current way missing values are handled is a bit convoluted, and probably a bit buggy

Edit: now documented in https://github.com/scikit-learn/scikit-learn/issues/32178

```
FAILED sklearn/ensemble/tests/test_forest.py::test_missing_values_is_resilient[make_regression-ExtraTreesRegressor] - ValueError: Unconsistent improvement 222739.5025534111 < 230516.5948817273
FAILED sklearn/ensemble/tests/test_forest.py::test_missing_values_is_resilient[make_classification-ExtraTreesClassifier] - ValueError: Unconsistent improvement -5.2 < -3.1999999999999993
FAILED sklearn/ensemble/tests/test_forest.py::test_missing_value_is_predictive[ExtraTreesRegressor] - ValueError: Unconsistent improvement 108.8032853873815 < 110.47401111474339
FAILED sklearn/ensemble/tests/test_forest.py::test_missing_value_is_predictive[ExtraTreesClassifier] - ValueError: Unconsistent improvement -14.078947368421051 < -11.323308270676694
FAILED sklearn/tree/tests/test_tree.py::test_missing_values_poisson[ExtraTreeRegressor] - ValueError: Unconsistent improvement 2961.5788534750745 < 2972.4751934414253
FAILED sklearn/tree/tests/test_tree.py::test_missing_values_is_resilience[42-None-make_friedman1-ExtraTreeRegressor-0.07] - ValueError: Unconsistent improvement 955.3035680542237 < 992.3319785056362
FAILED sklearn/tree/tests/test_tree.py::test_missing_values_is_resilience[42-None-make_friedman1_classification-ExtraTreeClassifier-0.12] - ValueError: Unconsistent improvement -5.083333333333333 < -4.0
FAILED sklearn/tree/tests/test_tree.py::test_missing_values_is_resilience[42-ones-make_friedman1-ExtraTreeRegressor-0.07] - ValueError: Unconsistent improvement 955.3035680542237 < 992.3319785056362
FAILED sklearn/tree/tests/test_tree.py::test_missing_values_is_resilience[42-ones-make_friedman1_classification-ExtraTreeClassifier-0.12] - ValueError: Unconsistent improvement -5.083333333333333 < -4.0
FAILED sklearn/tree/tests/test_tree.py::test_missing_value_is_predictive[42-ExtraTreeClassifier-0.53] - ValueError: 
```

On the branch for my PR, none of those tests fails.


### Versions

```shell
System:
    python: 3.12.11 (main, Aug 18 2025, 19:19:11) [Clang 20.1.4 ]
executable: /home/arthur/dev-perso/scikit-learn/sklearn-env/bin/python
   machine: Linux-6.14.0-29-generic-x86_64-with-glibc2.39

Python dependencies:
      sklearn: 1.8.dev0
          pip: None
   setuptools: 80.9.0
        numpy: 2.3.3
        scipy: 1.16.2
       Cython: 3.1.3
       pandas: None
   matplotlib: 3.10.6
       joblib: 1.5.2
threadpoolctl: 3.6.0

Built with OpenMP: True

threadpoolctl info:
       user_api: blas
   internal_api: openblas
    num_threads: 16
         prefix: libscipy_openblas
       filepath: /home/arthur/dev-perso/scikit-learn/sklearn-env/lib/python3.12/site-packages/numpy.libs/libscipy_openblas64_-8fb3d286.so
        version: 0.3.30
threading_layer: pthreads
   architecture: Haswell

       user_api: blas
   internal_api: openblas
    num_threads: 16
         prefix: libscipy_openblas
       filepath: /home/arthur/dev-perso/scikit-learn/sklearn-env/lib/python3.12/site-packages/scipy.libs/libscipy_openblas-b75cc656.so
        version: 0.3.29.dev
threading_layer: pthreads
   architecture: Haswell

       user_api: openmp
   internal_api: openmp
    num_threads: 16
         prefix: libgomp
       filepath: /usr/lib/x86_64-linux-gnu/libgomp.so.1.0.0
        version: None
```

COMMENT:
This issue and the two linked issues show the need for testing that the best split function does return the best split in all configurations. I think I have a good idea of how to do it. I will a open a PR with to add such a test, it's expected to fail for now, as long as the two PRs linked above aren't merged.

Labels: Bug, module:ensemble