If the airspeed indicator has inputs regarding air density (static port) and temperature. Why isn't Indicated airspeed from the indicator simply the same as True airspeed
Well, indicated airspeed is the most important speed a pilot needs to know, so it makes sense to have that in the most obvious position.
IAS can be thought of as the pressure that the wings are experiencing - therefore all of the stall, manoeuvring, flap extension, and landing speeds etc are based off indicated airspeed.
True airspeed is really only important for flight planning and navigation. Given that more and more aircraft have GPS nowadays to give the groundspeed, the need for the pilot to know the TAS in flight is further reduced.
In its basic form, the airspeed indicator measures pressure differential: between the dynamic pressure (from the pitot tube) and the static pressure (from the pressure port). As it only sees differential, it couldn't possibly tell "true" airspeed. That would require, as you say, factoring air density, which is usually not available to the pneumatic instrument.
From a practical point of view, the aircraft flies in terms of indicated air speed: in the sense that the indicated stall speed will be the same at all heights, where a speedometer showing true airspeed will require the pilot to make a calculation to know what the stall speed (or any other speed related to the movement of the airframe through the air) is. There is a single exception to this, as far as I know, which is the VNE: since the maximum speed is determined not by direct flying characteristics of the wing but by stress and vibration, it depends on true airspeed and not on the indicated one; so, at height, the pilot needs to be aware that the VNE shown in the airspeed indicator is not the real one. At the extreme, we have the "coffin corner", the height where the (true) stall speed coincides with the VNE, and it becomes impossible to fly the aircraft without either stalling it or destroying it.