Alternatives to imputation of missing values?
Supposedly you had these models, divided into subspaces, then what if in your test dataset a point is close proximity to where the missing point is in your training data. What would be the best solution? Have the average output of the two closest models? Is the output binary? And then would you have to weight each model, based on that proximity? Then I am thinking that many submodels could overfit on the subset of the data. Just some thoughts, which may be challenging to overcome.

Moreover, you could use other types of Classifiers which deal with missing data, like some decision trees implementations.

Finally, data imputation is a delicate art, and doing it right will leverage your model. For instance, you wouldn't want to impute all the missing data with the average as then you may change your data distribution. Knn techniques or ensemble regressions would be preferred.