This is stated in the Brahma Mahā-Purāṇa.
Although the "seven-births" parts are not explicitly stated (and the devotee ladder starts from Agni-deva), the rest of it, all matches with what Vedanta Desikan says in his work.
Chapter 121 - Manifestation of Māyā, Brahma-purāṇa
श्रृणुध्वं मुनिशार्दूलाः प्रवक्ष्याम्यनुपूर्वशः । यथा कृष्णे
भवेद्भक्तिः पुरुषस्य महाफला ॥ २२९.३ ॥
- Listen, O leading sages. I shall mention in due order how a person shall attain devotion to Viṣṇu with great benefit.
संसारेऽस्मिन्महाघोरे सर्वभूतभयावहे । महामोहकरे नॄणां नानादुः खशताकुले
॥ २२९.४ ॥
तिर्यग्योनिसहस्रेषु जायमानः पुनः पुनः । कथंचिल्लभते जन्म
देही मानुष्यकं द्विजाः ॥ २२९.५ ॥
- 4 & 5. The worldly existence is extremely terrible. It strikes terror into all living beings. It causes sorrow to men. Hundreds of
miseries abound in it. A soul is born again and again in thousands of
species of lower living organisms, O brahmins, and with great
difficulty it attains human birth.
मानुषत्वेऽपि विप्रत्वं विप्रत्वेऽपि विवेकिता । विवेकाद्धर्मबुद्धिस्तु
बुद्ध्या तु श्रेयसां ग्रहः ॥ २२९.६ ॥
- After human birth, he attains brahmin-hood, power of discrimination, a sense of piety, and happiness in successive births.
यावत्पापक्षयं पुंसां न भवेज्जन्मसंचितम् । तावन्न जायते भक्तिर्वासुदेवे
जगन्मये ॥ २२९.७ ॥
- Until the ultimate destruction of all sins of men accumulated in different births, devotion to Vāsudeva (Vishnu) who is identical with the universe, is
तस्माद्वक्ष्यामि भो विप्रा भक्तिः कृष्णे यथा भवेत् । अन्यदेवेषु या
भक्तिः पुरुषस्येह जायते ॥ २२९.८ ॥
कर्मणा मनसा वाचा तद्गतेनान्तरात्मना
। तेन तस्य भवेद्भक्तिर्यजने मुनिसत्तमाः ॥ २२९.९ ॥
- 8 & 9. Hence, O brahmins, I shall tell you how devotion to Viṣṇu is effected. A person may be devoted to other Devas mentally, verbally
and physically. His soul is directed to those gods. Thereby, O
excellent sages, he may become inclined to perform sacrifices.
स करोति ततो विप्रा भक्तिं चाग्नेः समाहितः । तुष्टे हुताशने तस्य
भक्तिर्भवति भास्करे ॥ २२९.१० ॥
- Then, O brahmins, with great concentration and purity of mind be begins to love the Fire-god (Agni-deva). When the fire-god is propitiated he
becomes a devotee of the Sun-god (Surya-deva).
पूजां करोति सततमादित्यस्य ततो द्विजाः । प्रसन्ने भास्करे तस्य
भक्तिर्भवति शंकरे ॥ २२९.११ ॥
- O brahmins, he worships the Sun-god (Surya-deva) regularly. When that god is pleased he becomes devoted to Śaṅkara (Shiva).
पूजां करोति विधिवत्स तु शंभोः प्रयत्नतः । तुष्टे त्रिलोचने तस्य
भक्तिर्भवति केशवे ॥ २२९.१२ ॥
- He performs the worship of Śambhu (Shiva) duly and strenuously. When the three-eyed god is satisfied he becomes devoted to Keśava (Vishnu).
संपूज्य तं जगन्नाथं वासुदेवाख्यमव्ययम् । ततो भुक्तिं च मुक्तिं च स
प्राप्नोति द्विजोत्तमाः ॥ २२९.१३ ॥
- By worshipping the imperishable lord of the universe named Vāsudeva (Vishnu), O excellent brahmins, he attains worldly pleasures and
English Translation by G. P. Bhatt
One may access the Gita Press Gorakhpur's abridged Hindi version of the above from here. Also, the alternate link for the English Translation from here.
- On a side note, a similar thematic narration, yet again from the Vaishnavite perspective, is also found in the Skanda Mahā-Purāṇa Verses 2.7.19.(13-26).
Further, similar "hierarchies" may exist for other sects' Supreme deities too; two examples out of many, being as follows -
- One instance of which is described in the Devī Bhāgavata-purāṇa Chapter 1, Book 5; establishing the hierarchy of -
[ Brahmā < Vishnu < Rudra < Devi ] in that order.
- Yet another similar hierarchy is found in the Kulārṇava Tantra (II. 7-8) too, which establishes the hierarchy as follows -
Vedācāra < Vaiṣṇavācāra < Śaivācāra < Dakṣinācāra < Vāmācāra < Siddhāntācāra < Kaulācāra