I have mostly seen pilots slowly pushing the throttle to toga during takeoff. But why? Why don't they just directly put the throttle to toga? Wouldn't that save runway length?
The engines may accelerate slightly differently, resulting in an asymmetric thrust. Spooling up slowly at first (normally to about 60% N1) prevents this. After that, you can accelerate up to full TO/GA (or press the button) without any significant asymmetry.
A pure engine-side viewpoint:
Engines don't really like their power level being changed quickly. Doing so, one increases the thermal and mechanical stress over the engine parts, as well as the probability of engine failure, unstable operation or flame-out. One doesn't want some of these things happening during take-off.
Aircraft engines are pretty much reliable these days, but one should not expend their luck quota for no good reason.
(These considerations are not limited to aviation, any seasoned truck driver does the same with the throttle pedal unless there is a compelling reason to act quickly.)
On smaller planes, the way it was explained to me was a mix of letting the engine come up to speed more easily as well as smoothly applying the left turn tendency from the inertial changes from the prop spin.
This left turn tendency is less of an issue on a dual prop plane with counter rotating props, but it's still a lot harsher on the engine and isn't really needed. If you need maximum power from the start of your take off roll, you can hold the brakes to keep from moving forward until at full power rather than forcing the engine to handle extreme changes.
Smoothly adding power allows the pilot to look at engine parameters as engine increases RPM/N2 and possibly avoid a failure at a critical time. With multi engine aircraft it also avoids a bunch of asymmetric thrust and a possible runway excursion. Our Falcon 2000LXS manual actually recommends advancing the power levers from idle to takeoff power over a 3 to 5 second period.
There is rarely only a single reason for something. In the aircraft I currently fly, there is a dynamic counterweight system in the engine. This system absorbs torsional vibration of the crankshaft in normal running. However, rapid power setting changes can lead to the counterweights "detuning" and causing significant damage to the engine, as well as potentially loss of power.
Of course, if the engine lacks a dynamic counterweight system, this reason does not apply. In general however, most engines will last longer if they are operated with care rather than making rapid power setting changes. For regular public transport, minimising the cost to the company is important for financial viability.
I think you're a tad confused... they don't. Pilots typically push it to full or almost full throttle the second they begin to accelerate for take off. The slow increase you feel is actually not a slow increase in thrust (the force applied to move the plane forward), but the rate at which the heavy plane full of people is accelerating (increasing in speed).
Please remember Newton's second law, which tells us acceleration is Net Force divided by Mass. Massive objects such as a heavy passenger jet (such as a 747, which weighs 183,500kg without any people on it!) would accelerate veeeerrrryyy slowly even with a large force added to move it forward.
We can also use, for example, a muscle car at a drag strip. Why does the driver in the competition slowly push down on the gas? Why doesn't he (or she) just floor it? Well, they're actually adding gas/'thrust'/force very quickly (they have to, to move their heavy cars the entire 1/4 mile so quickly!) but it takes time to take a heavy object and make it go from not moving to moving VERY fast (cars often exceed 100mph by the end of the 1/4 mile, and passenger jets average around 4 or 5 hundred km/h groundspeed).
Hope this helped!