First, I think that part of the source of this misunderstanding is a conflation between conceptual and intellectual minds. Maybe I misunderstand what you mean by intellectual mind. However, it is certain that for Gelugpas a conceptual understanding of emptiness is vital and this is arrived at through intellectual analysis. To conclude that for Gelugpas intellectual views are irrelevant because no object can withstand analysis is to misunderstand the Gelug approach.
smcj wrote:Developing the correct intellectual view of emptiness is Sutrayana practice.
and for Tsongkhapa the intellectual/conceptual understanding of emptiness that results from this practice along with some understanding of bodhicitta is the minimum necessary prerequisite for initiation into tantric practice.
smcj wrote:Specifically the Gelugpas go to great lengths to intellectually disprove the four possible logical scenarios for how something can abide.
However, to negate existence yet also not assert nonexistence or vice versa steps outside the scope of logic and is unacceptable to Tsongkhapa
Of course, when you analyse an object by way of the four extremes of production, or one or many, then the object is not found. However, this analysis for Gelugpas does not exclude the conventional existence of the object. Moreover, and here is the relevant distinction for this discussion, for Gelugpas it is okay to say that such an analysis "finds" (རྙེད་) emptiness - of course this emptiness cannot withstand analysis itself - and this is why Gelugs can say things like emptiness is an ultimate truth but exists conventionally, and attribute it qualities such as being permanent, and give definitions such as emptiness/ultimate truth is that object found by a valid cognizer analysing the ultimate...
smcj wrote:There is no wiggle room for the mind other than to see that the intellect is inadequate to the task. The intellect has checkmated itself! That's the whole point.
The point for the Gelugpas is that the intellect is up to the task! It is just that nothing exists inherently and so although it is up to the task of establishing inherent existence there is nothing for it to find. For if the intellect was not up to the task then they could not establish emptiness. What such analysis is not up to task for is negating conventional existence!
smcj wrote: The Gelugpas I've come across do not consider Sutrayana and Vajrayana at all the same. In terms of emptiness, it is not as if they take a different position when they do Vajrayana practice, but that since there is no intellectual answer, they proceed non-intellectually..
For Gelugpas Sutrayana and Vajrayana are not the same, but the distinction based on method - not the view of emptiness.
smcj wrote:The Sutra Mahamudra is the "look at your mind" type of thing. If ,when you are looking at your mind you find anything, that's the wrong answer. The Prasanghika view makes it clear that anything you find should be examined for an essence. That is why I say that Sutrayana Prasanghika view is compatible with and supportive of Sutrayana Mahamudra.
For Gelugpas the view arrived at is exactly the same - Prasangika. What is different are the techniques and object analysed.
smcj wrote:Vajrayana practices are activities, actions. Vajrayana completion stage practices (dissolving things into emptiness) do not require intellectual support. But they do require actions.
Gelug teachers instruct practitioners to turn to a conceptual understanding of emptiness even after/at the time of the dissolution until they have achieved a non-conceptual realisation. I will leave it there...