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"I exist" is not synthetic since the concept of "exist" is contained int he predicate "I".
"I exist" is not a priori since it requires my experience of I to be known.
The fact that "I exist" is not a priori does not necessitate that the statement is synthetic.
Have philosophers overlooked this huge fact or what?
'"I exist" is not a priori since it requires my experience of I to be known.'
If it requires experience in order to be known then, at least on Kant's account, it is not analytic either. Critique of Pure Reason, A7/ B11 :
'It would be absurd to ground an analytic judgment on experience, since I do not need to go beyond my concept at all in order to formulate the judgment, and therefore no testimony from experience for that' (Kant, 'Critique of Pure Reason', tr. P. Guyer, Cambridge : CUP, 2008, 142).
For Kant all analytic judgments are a priori and therefore cannot be a posteriori.
Now, you may well not feel constrained by Kant's analysis but you use his terms and, so far as I can see, his senses of the terms. Maybe you are right : ''I exist" is not a priori since it requires my experience of I to be known.'' But you need in that case a different analysis of terms to get the result that 'I exist' is analytic a posteriori.
Work on this further. You may be on to something. But you do need to work on it further.
Sri Ramana Maharshi said, the world is said to be a reflection in the mind as it does not remain in the absence of mind. If the universe is a reflection, there must be a real object known as the universe in order that it might be reflected in the mind. But, truly speaking, there is no objective universe. To explain it the dream illustration is set forth. The dream world is created with some mental impressions. They are called predispositions and latent tendencies. They are subtle. Just as a whole tree is contained potentially in a seed, so the world is in the mind from subtle predispositions and latent tendencies. Then it is asked: A seed is the product of the tree which must have existed once in order that it may be reproduced. So the world also must have been there some time. The answer is, No! There must have been several incarnations to gather the impressions (predispositions and latent tendencies) which are re-manifested in the present form. I must have existed before as I do now. Admitting the existence of the world, I must admit a seer who is no other than myself. When I seek the Self and abide as the Self there is no world to be seen. Reality is the seer only and certainly not the world. Such being the truth the man continues to argue on the basis of the reality of the world. Whoever asked him to accept a brief for the world?
Sri Ramana Maharshi also said the whole universe is full of life. Being cannot be other than consciousness. Otherwise you cannot say that you exist. Consciousness is the reality. When consciousness is associated with limitations you speak of self-consciousness, unconsciousness, sub-consciousness, super-consciousness, human-consciousness, dog-consciousness, tree-consciousness and so on.
That, which rises and falls, is the transient ‘I’ consciousness. Permanent consciousness is ‘I-I’ continuously. There is no difference between matter and spirit. All matter is energy; it is power or force. All are resolved in the Self and the Mind. Body is found because there is body consciousness, arising from ‘I-consciousness’ which again rises from consciousness. What you are now considering to be body-consciousness is due to superimposition.
There is only one consciousness, called also body-consciousness, Self-consciousness, etc. They are relative states of the same Absolute consciousness. Without consciousness, time and space do not exist. They appear in consciousness. The objects arise from where you (knower I) rise. Know the subject first and then question about the object. The subject comprehends the object also. What is not in you cannot appear outside. There is no difference like external and internal. Mind is consciousness with limitations. Unlimited consciousness takes on limitations and becomes the mind.
Your existence is known by your experience. Descartes argued that it is also logically certain that we exist. To deny that you exist is an empirical matter, but can be stated in logical terms, however absurd it is to deny you exist. A priori or analytic truths are truths that are only true by reason alone or logic alone. Claiming that it is logically certain we exist by reason and recourse to empirical experience alone, may be a defendable claim, but how do I know that? What is presupposed is that I know anything at all, and this is the infinite regress argument. I personally take the view that we can ask certain questions to find some type of foundational knowledge of the world, and build things up from there; however, the issue is that this reasoning is circular. It's not controversial that even things we consider logically certain depend on our assumptions and axioms we use. That said, to doubt we know anything at all is entirely incoherent, and as Wittgenstein quipped, "If you tried to doubt everything you would not get as far as doubting anything. The game of doubting itself presupposes certainty."
A reductio ad absurdist argument is used to justify that I exist, and if I want to claim I know a priori that I exist, knowing I exist entails someone, an agent, perceiving reality. When we talk about truth, we want truths that are independent of the perceiver, not truths that depend on you or I being conscious to perceive things, we want to know that logical truths are true whether we exist or not, in any alternative universe, in any possible world. That is how I see what truth is, and it is possible claiming that I exist based on my own self-doubt could be logically certain, but again this is circular and depends on preconceived notions of "reality" or "truth" or "existence".
It may give you comfort to say it is absurd to deny we exist, and it seems to be a logical guarantee that we do exist, similar to mathematical truth and logical certainties, such as nothing moving faster than the speed of light, no two things being both taller than each other, and a triangle having three angles and sides. If something is both knowable by referring to empirical reality alone and by logic alone, this is a contradiction, and why Kant originally ruled this type of knowledge out. However, Quine did point out, no such divide between analytic and synthetic exists, and there are some things that are both known by logical analysis and by empirical verification. So the long answer to your question is: No, Western analytic philosophers for example did not overlook this question, some have accepted the possibility of simultaneous analytic and synthetic truth. I would refer you to Quine's work, especially the essay "Two Dogmas of Empiricism" for a starting point about this.
In all, philosophers may respond to these questions by offering different epistemologies, such as coherentism, and a coherentist would claim that beliefs such as "I exist" are known only insofar as other beliefs cohere with this belief. A foundationalist would argue that the belief "I exist" forms the foundation of one's knowledge that your senses do not fail you and you can know what reality is truly like, building all of scientific knowledge and your own understanding of reality through sense perception. The problems that one runs into when choosing between these two different theories of knowledge includes the infinite regress argument and defending circular reasoning (if I know A is true because I define A to true, what justifies this belief — it's circular). There's not a clear answer, and often a pragmatic approach is used, as to which way of looking at knowledge is best or carries the most utility. Another interesting epistemology is one where knowledge and sense experience are one and the same, and there is no distinction between perception and knowledge, and the philosopher Wilfrid Sellars is very integral to that discussion.
