But there is no reason from an API perspective why it couldn't work like this. An ARIMA model is a handful of parameters that act on input and output vectors. Whether it makes sense to use it that way is a separate question.

As it happens, this is precisely how sktime works. The whole point is that its API is analogous to that of Scikit-learn. This is clearly demonstrated in the example code:

    y = load_airline()
    y_train, y_test = temporal_train_test_split(y)
    fh = np.arange(1, len(y_test) + 1)  # forecasting horizon
    forecaster = ThetaForecaster(sp=12)  # monthly seasonal periodicity
    forecaster.fit(y_train)
    y_pred = forecaster.predict(fh)

Sktime is just an implementation of various time series models with a Scikit-learn-compatible API. It is still up to the user to know what to do with this stuff.