TMingyur,TMingyur wrote: Now considering all this what is the use of specially emphasizing the "nature of the mind"?
The use is that the emptiness of the mind can be experienced right off, whereas the emptiness of the apple, for instance, can only be conceptualized right off. Beyond that, in a much shorter time than can emptiness of outer phenomena be realized, one can gain real experience of the emptiness of mind, of it's luminous nature, and of it's unobstructed capacity. Since when one gains actual realization of the nature of mind, not just glimpses and understanding, one also gains realization of everything else, it is a much swifter path for those suited to it. On the other hand, using analytical and conceptual methods takes much, much longer to realize. Analysis is a useful tool, but if one can use methods that allow for direct experience, that is obviously better. All 4 Tibetan traditions have these methods in addition to analytical ones.
To give a quote from a prayer by the late Dzogchen master Dudjom Rinpoche:
By means of the great weapon,
Indestructible primordial wakeful awareness,
May the void life-force of samsara and nirvana
Both be severed, at once.
Then, in the unending great bliss of Nyema’s feast
May we always enjoy the activity
That is beyond joining and parting.
In the pervasive space of evenness,
Even the word “suffering” does not exist—
So who ought to be striving for happiness?
In the Kingdom of Samantabhadra
Happiness and suffering are of one taste;
Without grasping, they liberate of themselves.
May I attain Samantabhadra’s kingdom in this very life!
This is much more easily realized by going directly to the nature of mind than it is by conceptualizing the emptiness of phenomena and meditating on that, which is why realizing the nature of mind is referred to as "knowing the one thing that frees all" in the Dzogchen tradition.