Could aircraft already do a safe auto-takeoff with todays equipment?
With some refinement and programming, the technology is easily there, sure; it just needs a few million dollars of design and certification costs for very little benefit. 

The main reason we have autolands is because it saves airlines money, by allowing the plane to land safely in minimal visibility instead of diverting to another airport. Aircraft taking off don't have this problem as they can do it in just about zero visibility. If there is no commercial incentive, the reality is it is not going to happen. Only when pilots are considered surplus to requirements will this ability be developed in earnest. 

There could be an argument made that it should be done for safety's sake, but I don't buy it. Pilot errors made on the takeoff run, resulting in an accident are incredibly rare. Maybe an auto takeoff could prevent a few tailstrikes but I can't see much more benefit.

Whenever you introduce something new you have to think long and hard about the ramifications of it. There's a whole discussion to be had on how rejected takeoffs would work - generally speaking, you do not want automation doing something that the pilots aren't expecting. It is sometimes better to take an issue into the air than take the risk of a high-speed RTO even under V1, how will the computer judge that when there are few hard and fast rules for it? Do you trust the human programmers to account for every single scenario? Then there's pilot competency, how do they keep their skills up when they could go months without flying the aircraft? If they are in the habit of delegating their decision making to the computer, will they be able to handle the heat in that career-defining moment when the computer fails and it's all on them?