upadekṣyanti te jñānaṁ jñāninas tattva-darśinaḥ
Know the ways of sacrifice [i.e., sādhanās] should be learned by submission, exhaustive inquiry and by service [i.e., obedience to instruction]. Those who have seen the truth are knowledgeable about the means, and they will instruct you if you have the above qualifications. (Gita 4.34)This instruction is given in relation to the series of yajñas listed in Gita 4.25-33. Yajña is used in the Gita as a synonym for sādhanā. One should take one's sādhanā from the guru. Mañjarī-bhāva is a sādhanā.
You don't get the siddhi, you get the sādhanā. A siddha-deha doesn't mean you are siddha. The sādhanā is performed internally with the siddha-deha. And that which you practice is the unripe form of what you achieve, which is the ripe form.
All India sādhanā traditions are oral as well as written. We tend to value the written traditions over the oral, but that would be a mistake. Paramparā means what is transmitted orally.
The siddha-deha is like the mantra. It is an initiatory process. You get the siddha-deha in order to practice rāgānugā bhakti. You are "initiated" into the practice. Just as in ancient times you would be initiated into a yajña. That is where the word dīkṣā comes from, historically.
So, I would say, being a little bold here, that people in Iskcon and the Gaudiya Math are not initiated, at least not in the way that the paramparā developed in the post-Chaitanya period. The difficulties in explaining the historical verity in Bhaktisiddhanta Saraswati's initiation, the idea of the śikṣā or bhāgavata paramparā and so on, are all clues to the lack of an initiation in the Gaudiya Math. But the strongest evidence of a lack of initiation is the outright rejection of the sādhanā of mañjarī-bhāva.
Of course, initiation is there. It would be ridiculous to say there isn't any initiation at all, since there is a ritual and an initiation into a particular process that looks very similar to the rest of the tradition, where mantras and so on are concerned. So it may seem very arrogant to say there is no initiation, but it seems fairly clear to me that the ideas of the siddha-deha being revealed from within, by the Holy Name, and other details of difference indicate the creation of a new or different initiatory tradition.
On the other hand, one may say it is not initiatory because it is exoteric rather than esoteric. Anything that is freely available to everyone is, by definition, not initiatory. Initiation means a special connection, or sambandha-viśeṣa. In fact, as I have been saying, the idea of sambandha-viśeṣa is not insignificant.
In Bhakti-sandarbha, Jiva Goswami, while talking about dāsya of the nine kinds of bhakti practice (nava-vidha-bhakti) says astu tāvat tad-bhajana-prayāsaḥ kevala-tādṛśatvābhimānenāpi siddhir bhavati. "Ultimately, there is no need for any effort in bhajan, since perfection is attained simply by having the identity of being a servant of God."
So the practice of mañjarī-bhāva-sādhanā is the cultivation of the identity of service in a particular mood, that of serving the perfect union of the Divine Couple. To be initiated by someone who has perfected that practice is the ideal and direct way to attain this identity. One should be wary of indirect means. If the Holy Name gives one the siddha-deha, as Bhaktivinoda Thakur says in kṛṣṇa nāma dhore koto bol, it is because the Holy Name guides one through the processes of sādhanā, citta-śuddhi sarva-bhakti-sādhana udgama (3.20.13).