The adjustment from Kṛttik?? to Aśvinī as starting point was probably to align the nakṣatras with the twelve-sign zodiac rather than with precession as such, but yes, it is perfectly possible that the counting originally started from Kṛttik?? because the conjunction of the sun (or, perhaps more likely, the full moon) with it coincided with some seasonally important event.

Indians usually use 27 Nakshatras, not 28. They start with Ashvini because they have long since shifted from their position in the Vedic era. Nakshatras have been sidereal from the very beginning. Therefore, an amendment was needed. But this does not change the essence of Nakshatras and the Hellenes did not play a role here. The very first list of nakshatras is in Yajurveda 4.4.10. It has 27 nakshatras. But whether they were unequal in size is unknown and completely unverifiable. All of today's India uses equal nakshatras, following the Lagadha pandit (5th century BC). The Namboodiri Brahmins (this is the oldest and ultra-orthodox Vedic Shrauta Shakha) use a standard system of 27 equal nakshatras. The ancient Vedic lists of Nakshatras began with Krittika, because this Nakshatra was occupied by the Vernal Equinox point. Later, the Vernal Equinox point was occupied by Bharani, Ashwini and even later Revati. And today the Vernal Equinox point is occupied by Uttara Bhadrapada. Therefore, now you can write lists of Nakshatras even from Uttara Bhadrapada. But this does not change anything at all, because the Nakshatras themselves are not tropical. Indian Nakshatras are tied to certain stars, and not at all to the moving point of the Spring Equinox. The Nakshatras are fixed in the starry sky, on the path of Chandra, without any point of the Vernal Equinox. The initial Nakshatra in any list is only a formality, therefore Varahamihira in his Brihat Samhita wrote lists of Nakshatras starting from different Nakshatras. Nakshatras are sectors of the ecliptic, and the ecliptic is the circle of the celestial sphere. The circle has no beginning at all.
Exactly 27 Nakshatras are the Hindu standard Janma-Kundali, based on the Yajur Veda Taittiriya Samhita 4.4.10, as well as on the Yajur Veda Kathaka Samhita 39.13.
The twenty-eighth Nakshatra Abhijit, whose lord is Brahman, was added in the Yajur Veda Maitrayani Samhita 2.13. But Brahman is not directly worshiped because it is a speculative concept. Therefore, Hindus usually do not take into account Abhijit Nakshatra. However, if someone is smarter than 1.161 billion traditional Hindus and even smarter than the ancient Indian astronomer and astrologer of Lagadha, well then let him use 28 Nakshatras for his health and let them be of different lengths or even tropical.
And regarding the Indian cult of Navagraha. Worship of the planets is already prescribed in the Atharva Veda 19.9.7,10. Scientific dating of this Veda: 12-9 centuries. BC. There are generally all nine Grahas in the Shruti, in the Kathaka Brahmana, as well as in the composite Navagraha Sukta. In Shruti there is only no Zodiac, therefore the zodiac Jyotish is Smriti-Jyotish. So, there is one main system of 27 Nakshatras, not two. Abhijit is known as the Nakshatra of Brahman, but is not used except for Muhurta and sometimes rare Dasha-systems. People are not born under Abhijit. Read Varahamihira, Garga, Parashara, and any of the classical Indian astrologers. The ancient Indians considered ζ Piscium to be the first degree of Ashwini. In all their ancient astronomical Sidhants (Surya S., Soma S., etc.) this is spelled out in black and white. Nakshatra Abhijit fell, according to Mahabharata 3.230.8-11 (4th century BC - 4th century AD).
Surya Siddhanta (4-5 centuries AD) divided the ecliptic into only 27 equal sectors.
Lagadha (4th century BC) did not use Abhijit Nakshatra.
Varahamihira (6th century AD) used Abhijit Nakshatra only for Muhurta, and did not use it as a birth-Nakshatra.
Lagadha and Varahamihira are the two main authorities of Indian astrology, historically reliable.

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