What fancy unencumbered by the laws? A curved distance is longer than s linear one.
You still want to knock out the BM in the boost phase, right?
Also, BMs don't travel in a curved path because those scientists who designed it didn't have the mathematical insight that you're propounding! The parabolic path is the most efficient.
Coming to the intercepting missile (in boost phase):
The boost phase is around 0-100 kms altitude from the launch point.
Assuming the launch is detected right at the time of BM launch, how long will be the reaction time? Go ahead and assume any value of your choice!
Then the intercepting missile needs to travel 200-300 kms laterally (assuming BMs are launched from deep inside) and an altitude of 50-100 kms - all this in time to intercept even though it had a lag of 'reaction time'.
How much faster should the interceptor missile travel in order to make up the longer distance in a shorter time?
Now if you attribute interceptor missile a velocity thrice that of the BM; To propel the missile at thrice the velocity the engine power needs to be around 3x, consuming 3x fuel rate! (x, y notation just for anecdotal notational purposes only. It's not accurate math)
If velocity is thrice that of BM then the drag associated would be 9 times as much as that of the BM (owing just to the velocity parameter). To overcome the additional 8y drag, the interceptor missile needs 8y more powerful engine consuming 8y more fuel.
So, cumulatively the missile needs 2x+8y more power engine consuming 2x+8y more fuel.
If the chosen 'reaction time' results in 4x veloctity requirement; then the missile will need 3x+15y more engine power consuming 3x+15y more fuel.
(Know that a lower altitude interception point will dramatically increase the velocity requirement as such pushing the weight & power attributes!)
All of this could make the interceptor missile size/weight comparable to the BM itself (assuming the weight owing to the larger engine & more fuel to be equivalent to the warhead of the BM).
Know that most of the BMs fuel is consumed in overcoming the gravity and drag in the initial stages of the trajectory - mid course has zero drag, and it almost coasts due to initial momentum!
Now that you have a fair idea of the size/weight of the interceptor missile, you should see how many of such missiles (missile batteries) are required along the border (closer to the border better the chances of interception). Also mull on how secure such large size missiles would be close to the border. Moving the missiles farther from the border will only require more powerful engine (and more fuel) to cover the additional distance in the short window for interception.
I intentionally left specific data & calculations out of the explanation - as an exercise to you, that might force you to think and understand.