There is some Deep Magic going on here. Let me try to offer a theory which, hopefully, will not muddy the waters further.
The ～て form of verbs (both positive and negative) implies a decision point. That is, at some point in time, you choose to do something or not to do something. Once this choice is made, it is irreversible.
Consider 食べてください. In effect, this request means, "At some point in the future, please choose to go from your current state of have-not-eaten to have-eaten." Once you enter the state of have-eaten, you cannot go back. You cannot undo the action of eating.
With ～なくて, a second action always exists, even if it is implied. That is, the decision is between not doing one thing and doing something else. 注意しなくて転んだ means that at a certain point, you chose not to enter the state of paying attention, and as a result you tripped. As with 食べてください, the decision (the choice of states) is treated as a single point in time and is irreversible.
This is why ～なくて cannot take ください. With 食べなくてください, you are essentially saying, "At some point in the future, please choose to go from your current state of have-not-eaten to have-not-eaten." This is illogical; you cannot change from state A to state B when A and B are the same. What you really want to say is, "Please stay in the state of have-not-eaten." But because ～て is limited to a single decision point, it cannot handle the range of points expressed in "stay in this state." To use an analogy from geometry, it is like the difference between a point (a discrete, finite quantity of one) and a line (an infinite quantity of points).
Because the action of staying in the state of have-not-eaten (the action of not entering the state of have-eaten) takes place over a range of time, the ～ないで form is needed. To explain why, consider what happens when we replace ください with いる. We find the same problem with ～なくて:
○ 食べている eating (or remaining in the state of have-eaten)
× 食べなくている (grammatically incorrect)
○ 食べないでいる not eating (or remaining in the state of have-not-eaten)
いる in these forms implies the continuation of a state. 食べている works because a state-change was made at some point (食べて) and you are continuing in that state (いる). 食べないでいる works because you started out in a state (食べないで) and are continuing in that state (いる). But 食べなくている does not work because there is no state-change in 食べなくて, therefore you cannot continue in it with いる.