It's not clear why you suggest that the $T_x$ component requires compensation.
In order to sustain circumnavigating motion a centripetal force must be provided.
When it comes to motion there is only one reference: the inertial coordinate system.
Granted: mathematical apparatus exists that allows you to use, say, a rotating coordinate system. Now: the equation of motion that you then use will contain a term for a centrifugal acceleration, commonly referred to as 'the centrifugal force'. That centrifugal term contains the angular velocity of the rotating coordinate system with respect to the inertial coordinate system.
There is no escaping the inertial coordinate system: you always use the inertial coordinate system as the reference of motion.
Since the inertial coordinate system is your reference of motion anyway the straightforward choice is to simply see it that way.
An unbalanced force causes acceleration. Here the $T_x$ component is providing a centripetal force, sustaining circumnavigating motion.
[Later edit] (in response to a comment)
There a psychological dimension here that needs to be recognized. It has to do with how perception of gravity works.
The perception of gravity is different from the perception of other ways force can be exerted upon you. Example: let's say someone is tugging your coat. Some parts of your coat will press against your skin, and you notice that.
Gravity however, acts on all parts of your body the same.
Perception of gravity builds up as follows: when you are standing up your feet have to carry the entire weight of your body, but your pelvis only has to carry your weight from the pelvis up; your neck only has to carry the weight of your head. All parts of your body have sensor cells that report how much compression they are sensing. From your ankle bones up to your neck bones there is a gradient in how much they are compressed. That directional gradient, combined with the sensor input of your equilibrium organ, gives rise to the perception of gravity.
That is: our perception of gravity is not a direct perception. Instead evolution has equipped us with a built in system to infer the presence of a gravitational force.
That system operates automatically. Wenever we sense this directional gradient of compression we infer the presence of a gravitational force.
In a car negotiating a bend
When you are sitting in a car, and the car is taking a corner a force towards the inside of the curve must be provided to sustain the curvilinear motion.
Your arm is against the door (of the car). The car is pushing against your arm, your arm is pushing against your torso. Your arm is the most compressed, followed by your torso, and your free arm is not compressed. So your physical sensation is one of a gradient of compression, with the compression increasing towards the part of your body that presses onto the door of the car.
That physical sensation is identical to being subjected to a centrifugal force because inertial mass is equivalent to gravitational mass.
This, I believe, is why everybody who first starts thinking about cirvilinear motion automatically starts with thinking in terms of a centrifugal force.
What is actually happening is that you are subjected to a centripetal acceleration, and because your body has inertia your body gets compressed a bit.