Stable extender contract via `fit` / `_fit` resp `predict` / `_predict` separation

### Describe the workflow you want to enable

tl;dr, I am suggestion to refactor `scikit-learn` internals to a layer separation with boilerplate between `fit` and `_fit` resp `predict` and `_predict` methods, to make extender interfaces more stable. Also see https://github.com/scikit-learn/scikit-learn/issues/31728

More background: Currently, every time `scikit-learn` releases a new minor version - e.g., 1.5.0, 1.6.0, 1.7.0 - compliant extensions, e.g., custom transformers, classifiers, etc, break - specifically, referring to the API conformance as tested through `check_estimator` or `parametrize_with_checks`.

These repeated breakages in the "extender contract" contrast the stability of the usage contract, which is stable and professionally managed.

For a package like `scikit-learn` which means to be a standard not just for ML algorithms but also an API standard that everyone uses, this is not a good state to be in - "do not break user code" is the maxim that gets broken for power users writing extensions.

Of course maintaining downwards compatibility is not always possible, but nothing should break without a proper warning.

### Describe your proposed solution

The `fit`/`_fit` separation would ensure stability of the extension contract - and would also allow to build secondary deprecation patterns in relation to it.

The (oop) pattern this would implement is the so-called "template pattern".

It would allow to remove likely changing parts such as the boilerplate (e.g., `validate_data` vs `_validate_data` and such) from the extension locus, and thus completely prevent breakage in relation to boilerplate changes.
Reference: https://refactoring.guru/design-patterns/template-method

Examples of how this can be used to improve stability:

* `sktime`, for a different API, has a separation between `fit` calling an internal `_fit`, where change-prone boilerplate is sandwiched between a stable user contract (`fit`) and a stable extender contract (`_fit`); similarly `predict` and `_predict`
* `feature-engine` overrides the `BaseTransformer` `scikit-learn` extension contract with a similar pattern using `super()` calls in `fit` etc.

The proposed `fit`/`_fit` pairing that combines strategy and template pattern can be introduced easily via pure internal refactoring - this also add a layer of tests that checks the stability of the extension contract.

As a side effect, this would likely reduce the amount of copy-pasted boilerplate among the core estimators in `scikit-learn` proper as well.

### Describe alternatives you've considered, if relevant

The obvious alternative is "do nothing" - which will lead to packages with extensions producing their own version of mitigating patterns for the unstable status quo.

The problem with this is that the extending packages are mutually incompatible with their extender pattern. It would be better if there is a single stable extension pattern prescribed - or at least recommended - by `scikit-learn`.

### Additional context

_No response_

COMMENT:
I think the point is not to force people to use this feature, rather they could. They can always override `fit` as they've been doing, and nothing changes in that regard.

As for validation logic, some of it can go into tags, and the rest we can start by letting developers implement themselves in their classes to start with.

```py
class BaseEstimator:
    def fit(self, X, y, **kwargs):
        validate_constructor_args(self)
        validate_args = self._validate_args()
        X, y = validate_data(X, y, **validate_args)
        if router(self):
            routed_params = process_routing(...)
        return self.__sklearn_fit__(X, y, **routed_params)

    def _validate_args(self):
        tags = get_tags(self)
        kwargs = {}
        if tags.input_tags.sparse:
            kwargs["accept_sparse"] = True

        ...
        return kwargs

    def __sklearn_fit__(self, X, y, **kwargs):
        ...  # algorithmic implementation here


class ProcessedEstimator(BaseEstimator):
    def __init__(self, param1, param2):
        self.param1 = param1
        self.param2 = param2

    def __sklearn_tags__(self):
        tags = super().__sklearn_tags__()
        tags.input_tags.sparse = True
        return tags

    def __sklearn_fit__(self, X, y, **kwargs):
        X, y = ...  # more validation here
        ...  # algorithmic implementation here

class RawEstimator(BaseEstimator):
    def __init__(self, param1, param2):
        self.param1 = param1
        self.param2 = param2

    def fit(self, X, y, **kwargs):
        self._validate_constructor_args()
        X, y = validate_data(X, y, accept_sparse=True)
        ...

        return self
```

Labels: RFC, Developer API