OP: Could it be argued that the thinker [the "I"] is, in essence, another thought?
Treating "the thinking" as Dasein, which does the thinking but also more (e.g. the living), the status of the "I" according to Heidegger is sketched out below. While able to be considered in thought, the "I" is posited as indeterminate, as a type or mode of being. Since Being founds Dasein and thoughts, Being is necessarily not a thought or thing itself. The "I" is part of Dasein's mode of being that "can embrace or own existence, or lose or disown existence," as in become lost to oneself in preoccupation, disengagement or inauthenticity.
From Being & Time, ¶ 25. An Approach to the Existential Question of the "Who" of Dasein
Perhaps when Dasein addresses itself in the way which is closest to
itself, it always says "I am this entity", and in the long run says
this loudest when it is 'not' this entity. Dasein is in each case
mine, and this is its constitution; but what if this should be the
very reason why, proximally and for the most [116] part, Dasein is
not itself? What if the aforementioned approach, starting with the
givenness of the "I" to Dasein itself, and with a rather patent
self-interpretation of Dasein, should lead the existential analytic,
as it were, into a pitfall? If that which is accessible by mere
"giving" can be determined, there is presumably an ontological
horizon for determining it; but what if this horizon should remain in
principle undetermined? It may well be that it is always ontically
correct to say of this entity that 'I' am it. Yet the ontological
analytic which makes use of such assertions must make certain
reservations about them in principle. The word 'I' is to be
understood only in the sense of a non-committal formal indicator,
indicating something which may perhaps reveal itself as its
'opposite' in some particular phenomenal context of Being. In that
case, the 'not-I' is by no means tantamount to an entity which
essentially lacks 'I-hood' ["Ichheit"], but is rather a definite kind
of Being which the 'I' itself possesses, such as having lost itself
[Selbstverlorenheit].
On this, William Blattner writes in The Cambridge Companion to Heidegger's Being and Time, ch. 14:
Heidegger is willing to grant that the word “I” designates in each
case the “owner” of existence, but he rejects the subjectivist or
egoistic philosophical baggage that the term often carries. The self
who I in each case am is not a sphere of subjectivity, a domain of
inwardness, or a field of self-consciousness. Rather, the “I” is in
each case simply that which can embrace or own existence, or lose or
disown existence.3 This contrast between owning and
disowning or losing existence is the “terminologically strict”
meaning of the terms “authentic ” and “inauthentic .”
...
3 Heidegger’s formulation also leaves open which existence is
mine. That is, on a traditional subjectivist understanding, I am the
field of experience that is in principle accessible to my
self-consciousness. Heidegger problematizes the “boundaries of the
self,” opening the door to potentially revolutionary reconfigurations
of our understanding of the self.
Returning to Being & Time, ¶ 25.
Yet even the positive Interpretation of Dasein which we have so far
given, already forbids us to start with the formal givenness of the
"I", if our purpose is to answer the question of the "who" in a way
which is phenomenally adequate. In clarifying Being-in-the-world we
have shown that a bare subject without a world never 'is' proximally,
nor is it ever given. And so in the end an isolated "I" without Others
is just as far from being proximally given. If, however, 'the Others'
already are there with us [mit da sind] in Being-in-the-world, and
if this is ascertained phenomenally, even this should not mislead us
into supposing that the ontological structure of what is thus
'given' is obvious, requiring no investigation. Our task is to make
visible phenomenally the species to which this Dasein-with in closest
everydayness belongs, and to Interpret it in a way which is
ontologically appropriate.
... If the 'I' is an Essential characteristic of Dasein, then
it is one which must be Interpreted existentially. In that case the
"Who?" is to be answered only by exhibiting phenomenally a definite
kind of Being which Dasein possesses. If in each case Dasein is its
Self only in existing, then the constancy of the Self no less than
the possibility of its 'failure to stand by itself'1
requires that we formulate the question existentially and
ontologically as the sole appropriate way of access to its
problematic.
But if the Self is conceived 'only' as a way of Being of this
entity, this seems tantamount to volatilizing the real 'core' of
Dasein. Any apprehensiveness however which one may have about this
gets its nourishment from the perverse assumption that the entity in
question has at bottom the kind of Being which belongs to something
present-at-hand, even if one is far from attributing to it the
solidity of an occurrent corporeal Thing. Yet man's 'substance' is
not spirit as a synthesis of soul and body; it is rather existence.
As the foundation of Dasein and beings, Being cannot be a present-at-hand thing, so the "I" cannot be a thought-thing.