Could an aircraft be tied down, then throttle up and release for takeoff in order to use a shorter runway?
As already mentioned, aircraft cannot take off without moving through the air, because it is the airflow around the wing that generates the lift required for take-off, and the only practical way of generating sufficient airflow around the wing is to move the wing through the air. This is typically done by moving the wing, and thus also the fuselage it is attached to, forward.

There is, of course, another way to look at that: aircraft can take off without moving relative to the ground, if exposed to a strong enough wind from the front of the aircraft.

That said, keeping the aircraft stationary, "throttling the motors full power, and then letting lose" is a very common short-field take-off technique at least for propeller aircraft; there's nothing theoretical about that. The aircraft is, however, typically kept stationary by applying full brakes while applying full engine power, rather than by using an external tie-down.

The tail likely isn't designed to take the force of holding the aircraft stationary while the engine is running at full whallop, although there's no reason why such an anchor point couldn't in theory be installed (see motor aircraft towing gliders to altitude, scale up somewhat, and you're close).

Since the brakes are already there and need to be powerful enough to at the very least hold the aircraft stationary during the engine run-up prior to take-off, which happens at a significant fraction of full power and can quite well happen at a hold short point just before the runway, there's no reason to spend weight on reinforcing the tail section to that point. Instead, just use the brakes for the few seconds needed.