Why can't we sample without replacement for each tree in a random forest if the subsample size is large enough?
No, the samples will not be independent, there is possibility the data samples will be skewed.
For example, imagine a class-imbalanced binary problem, once the minority class is already sampled (large possibility that this can happen given $n$ and $m$) then, without replacement, the rest trees will only sample from the majority class which will produce skewed trees.
Some references:

“Bagging based on resampling with and without replacement is equivalent”, is it?


For random forests, generally, the concept of replacement is
considered essential. This is because the underlying concept of random
forests is bagging to prevent overfitting, i.e., bagging builds an
ensemble of estimators trained on data with a high variance (with
regard to the training data they have seen).


Why does random forest use sampling with replacement instead of without replacement?


The basic idea of bootstrapping is that you use your sample as a
population. And from it you sample repeatedly, with replacement, to
build other samples of the same size as your original sample.
Replacement is an integral part of this process because you are trying
to create other possible sample distributions which could have come
for your original population based on the sample you have.