What the Guru Gita is saying should be taken more as a guideline for aspirants rather than as a test or a guideline to an aspirant's advancement. There is no stage at which an aspirant turns a Mauni. I have personally met a few sadhus who kept complete silence at all times. Have met several who have kept a vow of not speaking one day a week. Some have been very advanced sadhus, others were beginners.
The purpose of silence is austerity, control of the mind by controlling the tongue. When one is silent, one tries to be the Witness to Maya rather being a player within Maya.
Krishna says in the Gita (17.15) (Swami Nikhilananda translator):
Words that do not give offense and that are truthful, pleasant, and beneficial, and also the recitation of the Vedas--these are said to be the austerity of speech.
And Krishna says further in Gita (18.52) (Swami Gambhirananda translator):
One who resorts to solitude, eats sparingly, has speech, body, and mind under control, to whom meditation and concentration are ever the highest (duty), and who is possessed of dispassion;
And Sankara says in his work Aparoksanubhuti (108-109) (Swami Vimuktananda translator):
Who can describe That (i.e. Brahman) whence words turn away? (So silence is inevitable while describing Brahman). Or if the phenomenal world were to be described, even that is beyond words. This, to give an alternative definition, may also be termed silence known among the sages as congenital (inseparable from Brahman). The observance of silence by restraining speech, on the other hand, is ordained by the teachers of Brahman for the ignorant.
A good reference for the what the duties of an aspiring sannyasin are and what the characteristics of a knower of Brahman are is the Paramahamsa Upanishad.