I would alao believe that circumstances had attributed to the qualification of conventional existence. It was written in Je Rinpoche's Lam Rim Chen Mo:
One indication of a failure to recognise these critical points in not distinguishing in the beginning between those who are well-trained in the sutras or tantras and those who have no training at all, and then not assigning the appropriate amount of practice. Another such indication is that meditators are criticised if they study or do research. These mistaken customs persist in tibet.
We note that in the later parts of the same section, Je Rinpoche continues on to elaborate on how analytic meditation is required in order to achieve a fundamental understanding of the correct view. In the last paragraph of the section, he made the following statement:
Moreover, to claim that all conceptual thought involves apprehension of signs of true existence, and thus prevents enlightenment, is the worst possible misconception insofar as it disregards all discerning meditation. This is the system of the Chinese abbot Ha-shang.
With regards to this statement, i would probably infer that je rinpoche wanted to clarify and refute people who ardently believe that by practising non-conceptual mediation only, this would lead to enlightenment, which, i would put it in his terms, a form of nihilism. And perhaps because of so, he has started a tradition which does not negates conventional existence, but to attribute them to dependent arising, which in that sense (from my pov, don't quote me!) wouldn't be exactly wrong.
On the other hand, from what I gather, Mipham also criticises Je Rinpoche being an Autonomist, and it could be possible that Mipham attributes this to the acceptance of conventional existence as a characteristic of an autonomist. However, i wouldn't think that Gorampa's issue with Je Rinpoche primarily lies upon the characteristics of the autonomy and consequentialism, so that'd be another issue.
Tsongkhapa simply thought the refutation of the four extremes without qualification to be illogical.The non-conceptual mind is not the issue for Tsongkhapa here. The position Tsongkhapa is forwarding is that the conceptual mind using logical reasoning can conceptually realize emptiness and that this then acts as a bridge to non-conceptual wisdom.
I would think this is the case. Allow me to quote from (although controversial) Pabongkha's works that one has to train in analytic meditation, and through calm-abiding, unite analysis and calm abiding to achieve superior insight. I think the final point of uniting analysis with calm abiding is pretty much the same across all traditions, but something unique about this tradition would be the training of analytic mediation, right from the start. Again i quote from Pabongkha's Liberation:
Your meditations are analytic until you achieve mental quiescence.
Furthermore, in the way Je rinpoche refutes inherent existence via a consequentialist position (reducto ad absurdum), it would be noted that the object of refutation need to be precise, as we might fall into a trap of negating something that is not equivalent to the object of refutation, i.e. A class of objects more than or less than the object of refutation. I think it is with this in mind that Je Rinpoche gives a specific qualification of the refutation. (My opinion only though, it might not be right!)
Hope this helps!
Homage to the Mother of Buddhas as well as of the groups of Hearers and Bodhisattvas
which through knowledge of all leads Hearers seeking pacification to thorough peace
And which through knowledge of paths causes those helping transmigrators to achieve the welfare of the world,
And through possession of which the Subduers set forth these varieties endowed with all aspects.
- Ornament of Clear Realisation