The key to the weather stripping is to keep air from penetrating, if the new stripping fills the gap snugly, it should be fine.
The trick is to get it snug without impairing the operation of the door.
Remove the old weatherstripping completely and get a clean surface to attach the new stuff. The Sawzall may be overkill for the staples, try a small flat head screwdriver and some needle nose pliers.
Also inspect the sweep under the door.
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'Tis better to remain silent and be thought a fool, than open one's mouth and remove all doubt.
Samuel Johnson (1709 - 1784)
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