What does RandomForestClassifier() do if we choose bootstrap = False?
According to the definition in this link
bootstrap : boolean, optional (default=True) Whether bootstrap samples
are used when building trees.
Bootstrap = True
Bootstrap = False
It seems like you're conflating the bootstrap of your observations with the sampling of your features. An Introduction to Statistical Learning provides a really good introduction to Random Forests.
The benefit of random forests comes from its creating a large variety of trees by sampling both observations and features.
Bootstrap = False is telling it to sample observations with or without replacement - it should still sample when it's False, just without replacement.
You tell it what share of features you want to sample by setting
max_features, either to a share of the features or just an integer number (and this is something that you would typically tune to find the best parameter for).
It will be fine that you're not going to have every day when you're building each tree - that's where the value of RF comes from. Each individual tree will be a pretty bad predictor, but when you average together the predictions from hundreds or thousands of trees you'll (probably) end up with a good model.