Is it possible for an ordinary individual possessing sound cognitive faculties and access to publicly available evidence to establish the existence of a God? Is there a prevailing consensus in philosophy regarding this question?
Two disciplines that suggest that God's existence can be established through reason, arguments, and evidence that anyone should be able to inspect and study by themselves are natural theology and apologetics, but my impression is that not everyone would agree with this suggestion.
Note: When I mention publicly accessible evidence, I am referring to evidence that is in principle available to anyone. Additionally, a proper examination of this evidence, with sound cognitive faculties, should ideally lead to substantial degrees of intersubjective agreement.
Related questions
Are most philosophers atheists, monotheists, polytheists or what?
Is the teleological argument for God completely refuted?
Can God make the belief in His own existence justified (if He exists)?
Definitions
In the comments section definitions of evidence and God have been requested. I will provide quotes from a few sources as working definitions of these concepts.
Evidence
The concept of evidence is crucial to epistemology and the philosophy of science. In epistemology, evidence is often taken to be relevant to justified belief, where the latter, in turn, is typically thought to be necessary for knowledge. Arguably, then, an understanding of evidence is vital for appreciating the two dominant objects of epistemological concern, namely, knowledge and justified belief. In the philosophy of science, evidence is taken to be what confirms or refutes scientific theories, and thereby constitutes our grounds for rationally deciding between competing pictures of the world. In view of this, an understanding of evidence would be indispensable for comprehending the proper functioning of the scientific enterprise.
For these reasons and others, a philosophical appreciation of evidence becomes pressing. Section 1 examines what might be called the nature of evidence. It considers the theoretical roles that evidence plays, with a view towards determining what sort of entity evidence can be—an experience, a proposition, an object, and so on. In doing so, it also considers the extent to which evidence is implicated in justified belief (and by extension, knowledge, if knowledge requires justified belief). Then, section 2 considers the evidential relationship, or the relation between two things by virtue of which one counts as evidence for the other; and it explores the nature of their relationship, that is, whether the relationship is deductive, explanatory, or probabilistic. Finally, equipped with this theoretical background, section 3 looks at some of the important problems and paradoxes that have occupied those working in the theory of evidence.
(source: https://iep.utm.edu/evidence/)
God
God and Other Ultimates
What it takes to be ultimate is to be the most fundamentally real, valuable or fulfilling among all that there is or could be. Historically, philosophy of religion in the West has taken God to be ultimate. Over the past century, the field has become increasingly aware that ultimacy is grasped under different concepts in the world’s religions, philosophies and quasi-religious philosophies—so not only as “God” but also as, e.g., “Brahman”, “the Dao”, and more. Moreover, people have thought to conceptualize each of these ultimates in numerous ways across cultures and times, so there are many models of Brahman, many models of God, many models of the Dao, and more; perhaps there is even a model of what is ultimate for each person who has thought hard about it. This entry presents a framework for understanding this vast landscape of models of God and other ultimates and then surveys some of its major sights. Familiarity with this landscape can clarify the long journey to deciding whether there is anything ultimate, among other benefits.
Section 1 defines “ultimate” and “models” of ultimates, discusses reasons to be interested in the project of modeling what is ultimate or alternatively to think it futile, and explains major categories that help organize the field of models. Section 2 uses these categories to relay over twenty models of Brahman, God and the Dao, both for their own sakes and as entrées into the landscape (the models are numbered as they surface to help the reader spot them and to show by example what a model is). Section 3 discusses the significance of the plurality of models once they are juxtaposed.
[...]
The most venerable model of God that is often read dualistically is known as “perfect being theology”, which bears traces of its origin in its name (this is model 8—a general model, species coming). The idea fully grown, as we have it today, defines God as that which is perfect (whether personal or not), where perfection is typically taken to entail being unsurpassable in power, knowledge, and goodness, and several models add being immutable, impassible, a se, eternal, simple and necessary in some sense. Most perfect being theologians take God to have created the universe out of nothing (ex nihilo), and that view can be taken to entail dualism for a variety of reasons. To offer one, as Brian Davies says, “God makes things to be, but not out of anything” (italics his), including not out of Godself, so the cosmos is entirely fresh stuff—a second kind of stuff, distinct from and radically dependent on God (2004: 3).