"I" or "Me" in Bhagavad Gita is interpreted differently by different philosophers. But in Anugita Parva of Aswamedha Parva of Mahabharata clearly says Lord Krishna was in union with Supreme Brahman at time of Bhagavad Gita recitation. Different schools interpret that "Supreme Brahman" differently.
So, not only Vaishnava acharyas, even Advaitins believe in entire Bhagavad Gita. It was indeed Adishankara of Advaita Sampradaya who made Bhagavad Gita popular by writing commentary on it. Abhinavagupta of Kashmir Shaiva (Trika) also wrote commentary of Bhagavadgita.
Krishna says following in Anugita when Arjuna asked Him to repeat Gita.
'Vasudeva said, 'I made thee listen to truths that are regarded as
mysteries. I imparted to thee truths that are eternal. Verily, I
discoursed to thee on Religion in its true form and on all the eternal
regions. It is exceedingly disagreeable to me to learn that thou didst
not, from folly, receive what I imparted. The recollection of all that
I told thee on that occasion will not come to me now. Without doubt, O
son of Pandu, thou art destitute of faith and thy understanding is not
good. It is impossible for me, O Dhananjaya, to repeat, in detail, all
that I said on that occasion. That Dharma (about which I discoursed
to thee then) is more than sufficient for understanding Brahman. I
cannot discourse on it again in detail. I discoursed to thee on
Supreme Brahman, having concentrated myself in Yoga. I shalt now,
however, recite to thee an old history upon the same topic. O foremost
of all persons, observant of duty, listen to everything I now say, so
that, with an understanding adapted to my teaching, thou mayst succeed
in attaining to the highest end. O chastiser of foes, on one occasion,
a Brahmana came to us from the regions of Heaven. Of irresistible
energy, he came from the regions of the Grandsire. He was duly
reverenced by us. Listen. O son of Pritha, without yielding to
scruples of any kind, to what he, O chief of Bharata's race, said, in
answer to our enquiries, agreeably to heavenly forms.'
Adishankara of Advaita Sampradaya says "Vasudeva" in verse 7.19 means "innermost self" or "pratyagatman". This is what Adi Shankara says (translated by Alladi Mahadeva Sastri)
- At the end of many births, the man Of wisdom comes to me, (realising) that Vasudeva is the all : he is the noble-souled
(Mahatman), very hard to find.
Commentary: At the end of many births occupied in spiritual
regeneration as preparatory to the attainment of wisdom, the man of
mature wisdom resorts to Me, Vasudeva. the innermost Self
(Pratyagatman) —Realising that Vasudeva is the All. He who thus comes
to Me, Narayana, the Self of All, is a Mahatman, a man of high
soul ; there is no other either equal to him or superior to him.
Therefore such aman is very hard to find ; it has indeed been said
that s' among thousands of men, one perchance strives for perfection
" (vii. 3.)
Advaita Sampradaya emphasize on realizing our inner self (Atman) as ultimate reality.