You have things backwards.

A sinking/long-travel pedal means you have excessive air in the line. Because the air compresses, you CANNOT develop full braking pressure. The booster has nothing to do with this.

If the booster is leaking or out of the circuit entirely, you will have a hard brake pedal. You can still stop the car like this, but it takes extra effort. It is designed so that if you lose vacuum or the booster fails that you can still safely stop the car.

Get the air out of the system FIRST. With the engine stopped and the vacuum bled from the booster, your brake pedal should be nearly rock hard. It shouldn't take much -if any- travel to encounter significant resistance. If it is spongy or takes the 2" you claim it does to get startlingly firm, you have more bleeding to do.

Fix the air in the system FIRST. It is a serious safety issue!

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