>enough time to react and fire
You mention the intercept window, but it's more complex than interceptable / not interceptable, and time is not the only factor at play.
It's much easier to intercept the missile earlier rather than later, even if the window allows for a later intercept.
It's easiest during ascent phase, because: 1) the rocket is still accelerating (slower target), 2) the rocket is still burning (bright target) 3) before the payload(s) has separated from the booster (big target) 4) before warheads and decoys deploy (single target)
There are some other aspects to consider as well, like a trajectory-changing near-miss during early ascent will have a much greater impact than the same in late ascent, (depending a lot on the target type).
Which is why US missile defense began as an entirely ship based program, to be deployed where ascent interception would be possible.
Although this has changed a great deal since then; technical abilities and limits revealed through testing, as well as political/ budget constraints have shifted the tactical role of the system more state-side, not without controversy.