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.