Why does ISKCON emphasize bhakti yoga over karma and gyan yoga?Which Yoga does the Vedas most emphasize?
Partial answer to:
Q. Why does ISKCON emphasize bhakti yoga?
It's because ISKCON subscribes to Gaudiya Vaishnava Sampradaya established by Chaitanya Mahaprabhu.
Pure devotion to Lord Krishna is the most purposeful practice of the jivas according to Śikṣāṣṭakam, the only composition written by Sri Chaitanya Mahaprabhu himself.
Quoting a verse from shikShAShTaka:
न-धनं न-जनं न-सुन्दरीम् कवितां वा जगदीश कामये ।
मम जन्मनि जन्मनि ईश्वरे भवताद् भक्तिः अहैतुकी त्वयि ॥ ४॥
O almighty Lord, I have no desire to accumulate wealth, nor do I desire beautiful women, nor do I want any number of followers. I only want Your causeless devotional service birth after birth.
Another point is: one of the mission of ISKCON is "To propagate a consciousness of Krishna as it is revealed in the Bhagavad-gita and Srimad Bhagavatam"
And Lord Krishna instructed Arjuna "the most confidential knowledge (refer BG 18.64)" in last chapter of Bhagavad Gita:
TEXT 65: Always think of Me, become My devotee, worship Me and offer your homage unto Me. Thus you will come to Me without fail. I promise you this because you are My very dear friend.
ISKCON belongs to Gaudiya Vaishnava tradition and is one of its branches.
Gaudiya Vaishnava Theology is primarily based on Bhagavata Purana, Bhagavad Gita, Gopala Tapani Upanishad, Vishnu Purana which all emphasize bhakti.
For eg: Bhagavata Purana says this: Before Vyasa compiled Bhagavatam, he sat in meditation and observed that bhakti puts ends to all miseries.
SB 1.7.3 — In that place, Śrīla Vyāsadeva, in his own āśrama, which was surrounded by berry trees, sat down to meditate after touching water for purification.
SB 1.7.4 — Thus he fixed his mind, perfectly engaging it by linking it in devotional service [bhakti-yoga] without any tinge of materialism, and thus he saw the Absolute Personality of Godhead along with His external energy, which was under full control.
SB 1.7.5 — Due to this external energy, the living entity, although transcendental to the three modes of material nature, thinks of himself as a material product and thus undergoes the reactions of material miseries.
SB 1.7.6 — The material miseries of the living entity, which are superfluous to him, can be directly mitigated by the linking process of devotional service. But the mass of people do not know this, and therefore the learned Vyāsadeva compiled this Vedic literature, which is in relation to the Supreme Truth.
Though Bhagavata Purana discusses about other yoga processes as well, primary aim as can be seen from above is to teach bhakti.
Even Sukadeva, who was himself a perfect jnani before, at the beginning of the narration to king Parikshit states this:
SB 2.1.7 — O King Parīkṣit, mainly the topmost transcendentalists, who are above the regulative principles and restrictions, take pleasure in describing the glories of the Lord.
SB 2.1.8 — At the end of the Dvāpara-yuga, I studied this great supplement of Vedic literature named Śrīmad-Bhāgavatam, which is equal to all the Vedas, from my father, Śrīla Dvaipāyana Vyāsadeva.
SB 2.1.9 — O saintly King, I was certainly situated perfectly in transcendence, yet I was still attracted by the delineation of the pastimes of the Lord, who is described by enlightened verses.
SB 2.1.10 — That very Śrīmad-Bhāgavatam I shall recite before you because you are the most sincere devotee of Lord Kṛṣṇa. One who gives full attention and respect to hearing Śrīmad-Bhāgavatam achieves unflinching faith in the Supreme Lord, the giver of salvation.
This is the conclusion of Brahma after reading Vedas 3 times(at the beginning of creation)
SB 2.2.33 — For those who are wandering in the material universe, there is no more auspicious means of deliverance than what is aimed at in the direct devotional service of Lord Kṛṣṇa.
SB 2.2.34 — The great personality Brahmā, with great attention and concentration of the mind, studied the Vedas three times, and after scrutinizingly examining them, he ascertained that attraction for the Supreme Personality of Godhead Śrī Kṛṣṇa is the highest perfection of religion.
Probably first two times he would have concluded karma and jnana as the conclusion of the vedas.
Also, in 15th chapter of Gita, Krishna gives the same understanding.
First he states that he is the knower of the Vedas.
Bg 15.15 — I am seated in everyone’s heart, and from Me come remembrance, knowledge and forgetfulness. By all the Vedas, I am to be known. Indeed, I am the compiler of Vedānta, and I am the knower of the Vedas.
After this he speaks the essence of Vedas in 5 verses.
The last two verses say that bhakti is the most confidential teaching of the Vedas.
Bg 15.19 — Whoever knows Me as the Supreme Personality of Godhead, without doubting, is the knower of everything. He therefore engages himself in full devotional service to Me, O son of Bharata.
Bg 15.20 — This is the most conﬁdential part of the Vedic scriptures, O sinless one, and it is disclosed now by Me. Whoever understands this will become wise, and his endeavors will know perfection.
Krishna reveals the same knowledge at the end of Bhagavad Gita:
Bg 18.64 — Because you are My very dear friend, I am speaking to you My supreme instruction, the most conﬁdential knowledge of all. Hear this from Me, for it is for your beneﬁt.
Bg 18.65 — Always think of Me, become My devotee, worship Me and offer your homage unto Me. Thus you will come to Me without fail. I promise you this because you are My very dear friend.
Bg 18.66 — Abandon all varieties of religion and just surrender unto Me. I shall deliver you from all sinful reactions. Do not fear.
Question: Which Yoga does the Vedas most emphasize?
Vedas talks about Karma Yoga and Gyan Yoga. Bhakti was not part of the Vedic religion. It got assimilated in a later period. For details see the following link in Hinduism stack exchange - Historical origin of Bhakti