I don't think torque is constant. The power is constant, and torque gradually decays as RPM increases as per the following equation:

HP = Torque x RPM ÷ 5252

Most 1000 HP drag cars are in their power band from the time the driver lets go of the trans brake until the race is over (unless it's a stick shift, but most drag cars are automatic).

Like any other conventional automatic, there is a torque converter between the engine and the transmission which allows the engine to spin faster than the transmission input shaft.

At the starting line, the driver engages the transmission brake, which locks the transmission and allows him to floor the engine, which brings it up to the optimal RPM (and spools the turbos if so equipped). It's almost the same as if you were to hold the brake and floor the accelerator at a red light. The only difference is that the brakes on a drag car wouldn't be able to hold it back, so they use the transmission instead.

Then, when it's time to start, he lets go of the trans brake and the power is instantly delivered to the wheels.

There usually isn't an issue with not having enough power at the starting line. It's actually the opposite. High power cars usually have to limit their starting RPM to avoid doing a wheelie or losing traction.