The thing is that net force of $0$ does not mean "no forces act on you". Indeed, you feel pushed into your chair as you sit on it, and if you were sitting in your chair on an upward accelerating elevator you would feel pushed into your chair even more. In either case the net force acting on you in your non-inertial frame is $0$ (once you take pseudo-forces into account), but you would "feel" more in the case of the upward accelerating elevator compared to just sitting in a chair on the ground.
This shows us the key to your question. We "feel in the opposite direction of acceleration (as observed from an inertial frame)." In the case of the elevator, we more we accelerate upwards, the more we feel pushed downwards. In circular motion, the acceleration is inwards, so we feel "pulled" outwards.
You can essentially link this to Einstein's equivalence principle: experiencing an acceleration is just like being in a gravitational field pointing in the opposite direction as the acceleration. Indeed, this is even why you feel pulled to the ground right now, you could argue that you are actually being accelerated upwards (approximately).
Another way to think about it. In non-inertial frames Newton's laws no longer hold. But which law/laws does/do not hold? Well, technically speaking Newton's second law no longer holds. In other words, we can feel forces without experiencing accelerations in our non-inertial frame. So it should be no surprise that we still feel an outwards force when we don't observe acceleration in our frame, if we felt nothing this law would hold!
However, this idea doesn't sit well with us. What if we still want Newton's second law to hold? Well then we introduce the idea of fictitious forces. However, by doing this we have now thrown Newton's third law out the window. Forces now longer have to have an equal and opposite partner. Forces do not have to arise from interactions anymore. So your claim that the centrifugal force is a reaction to the centripetal force is actually false! Remember, action-reaction pairs do not act on the same object. The centrifugal force arises from the non-inertial reference frame.
So I would argue that while we have introduced fictitious forces to help us keep N2L valid, I think it is better to say that we are "feeling" the effects of being in a non-inertial reference frame.