Source of confusion: Harry isn't dead. Why hasn't the Mysterious Enemy (ME) had him killed?

1) Harry isn't a target: he's just in the way of some other objective.

Let's lay that theory aside for the moment, on the grounds that it's not fun. The next most likely is:

2) ME's objective is to make Harry behave in a particular way.

So Harry is intended to go huge and dramatic. But that's a somewhat stupid plan on the part of ME, too prone to random factors derailing it. Except that we've already seen plans like that working, in one particular case: where time-turners are involved, and the person is manipulating themselves.

The easier way for Harry to achieve godhood is to break the restrictions on the time-turners, and go back arbitrarily far. This is fine - but the time-travelling future!Harry has to ensure that current!Harry forms the firm resolution to achieve godhood. So the ME is a time-travelling future!Harry, working to ensure that current!Harry does indeed follow the path to making the future version come into existence.

If we want to go all Ouroboros, we could have future!Harry influencing Harry's upbringing, feeding his "dark side" to ensure he ends up sufficiently intelligent and motivated to carry the plan through...