This is an very nice and interesting piece and I would like to initially address the idea of thought as a mental intrusion. I feel that when there is 'presence', that is when we are in a state of being fully aware in the present, then there are no intrusions. Letting go has become so natural that even a thought is allowed to arise and fall with ease and grace as part of this natural state. There is no stickiness to it anymore, nothing to cling to, and therefor nothing to label.
Thoughts may feel very coarse, and even shocking, when they arise from a state of stillness - but this feeling of coarseness may be due to our attachment to a pleasant state of mind - but this thought has also arisen in the present moment, has it not? And we can keep things simple by allowing it to be 'as it is' which is part of this natural state. If it was not part of this natural state then why would it arise? "Well I must have messed up something somewhere. Maybe I did something wrong. I can't seem to meditate without these thoughts arising." Perhaps the point is obvious. This labeling is the crux here, not thought or thinking, not distractions as these naturally fall away in this process. This critical discrimination is the opposite of discernment and insight. It is a feeding of the initial distraction. It arises ‘as it is’ and then we feed it and give it power, a reality, an identity.
Generally we (that is you and I:) wish to hold on to a pleasant state (and even mistake this pleasant state as something important) so naturally any other state that arises causes our aversion to arise. This aversion was already present in the form of our attachment to the pleasant feeling we have.
Awareness of the present is not pleasant or unpleasant, good or bad, right or wrong as these have dissolved into what simply ‘is’. There is a space of freedom, and of healing, where all is understood and welcomed as humanity, as life, yet we are not caught because we are not deceived.