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:
> Rather than further promoting your solution (split `fit` / `_fit`)

I was just directly responding to @GaelVaroquaux, I hope that is ok.

> it would help much more if you could provide the concrete uses cases that break your code. Which private functions do you need and why? For example, you mention `validate_data`, but I don't get what the problem is/was?

some exemplars for problems were listed here: https://github.com/scikit-learn/scikit-learn/issues/31912#issuecomment-3182305064

The generic use case is keeping a third party estimator up to date and passing `parametrize_with_checks`, and staying compatible when new `scikit-learn` versions are released.

The crux was, up until recently, it was entirely impossible to comply with the API contract implied by `parametrize_with_checks` without using private methods, and it currently still is.

Methods such as `_validate_data` or `_check_method_params` are (or were) necessary, and since they are private, 3rd party estimators can break contract without notice.

More concretely, suppose you are one of the 10 billion people who have their own `scikit-learn` compatible estimators in an open or closed code base, and want to ensure they remain compatible. Every minor release will break this compatibility, or at least has done so for at least the last three minor releases. Compare this to the "clean" deprecation messaging in the public interfaces, e.g., of `fit` or `predict`.

Labels: RFC, Developer API