What is philosophical argumentation? What is philosophy?
I'd define philosophy and philosophical argumentation roughly as the formalisation of how we figure out anything about anything.
Let's say you've figured out that every man seems to be mortal, and you meet a man named Socrates. You would just naturally conclude that Socrates is mortal. This is something we do already, but philosophy just formalises that into a syllogism, and this formalisation also helps us identify flaws in our intuition that could lead us to incorrect or unjustified conclusions. Although we use a lot of abductive or inductive reasoning in day-to-day life, and that isn't quite as concrete as deductive syllogisms, although we can use deduction to extend or test abductive or inductive conclusions. Taking the example above, we may not deductively know that every man is mortal, nor that Socrates is a man - these are merely the "best" conclusions we came to, based on the evidence. But we can still use those as premises in a deductive argument, to conclude something else.
Similarly, I'd define science roughly as the formalisation of how we figure out anything about reality. If you poke a bear with a stick and it bites you, and you poke another bear with a stick and it bites you, you may conclude that poking bears with sticks leads to them biting you (disclaimer: don't poke bears with sticks). Science is that exact same line of thinking, of trying to figure out cause and effect, but it's just putting it into a formal framework.
So these are very useful tools that allow us to understand why we believe what we believe and to refine and increase your knowledge.
Where I feel philosophers often go wrong, is to head off into Speculationville and Wordsaladtropolis, i.e. ending up with some conclusion by just stacking a bunch of premises on top of one another, or using a bunch of big words, even though the substance of what they're saying disappears when you break it down into plain English, or what they're saying is just so far removed from reality that you'd be forgiven for putting that in the Fiction section in the library.
So what about God and grounding with personal experience?
I'd ask a somewhat analogous question: Can belief in the existence of your spouse be grounded in (and justified by) personal experience rather than philosophical argumentation?
I'd trivially answer "yes". It's usually the theists who offer philosophical arguments, e.g. cosmological arguments, teleological arguments, ontological argument, moral arguments, transcendental arguments, presuppositional arguments, whereas atheists point out problems with these arguments (and a theist might see all these arguments and say "wow, there are so many ways to conclude that God exists", whereas an atheist would say "theists are just grasping at anything they can to try to justify their unjustified belief" and "a dozen bad arguments doesn't make for a good argument"). Although atheists do also have their own arguments against specific theist claims, e.g. the problem of evil.
However, for personal experiences, many people disagree about the "correct" conclusion to draw from so-called spiritual experiences. Some theists and atheists certainly draw different conclusions from similar experiences, but theists from different religions or different denominations of the same religion may also draw different conclusions about which deity or deities exist. So that's a massive issue with just trusting one's experiences without seriously questioning the correct conclusion to draw from them.
Imagine if you believed some specific human exists, but other people disagree. To resolve that, I wouldn't dig into philosophical arguments, but rather we should discuss what believed experiences you've had with them and what the reasonable conclusion is to draw from that.
This is something that people do discuss, and should probably discuss more than philosophical arguments.* Although discussions about how to interpret personal experiences also clearly haven't convinced everyone one way or the other either.
My answer to one of your earlier questions may be relevant: What exactly would count as a "positively meaningful and reciprocal conscious relationship" between a person and a God?
* I'll always be reminded of William Lane Craig pointing out that he's convinced by "the witness of the holy spirit", not by the philosophical arguments he promotes (and he thinks we should specifically lower our epistemic standards when it comes to Christianity). I'm not necessarily saying promoting an argument you aren't convinced by is a problem (a cynic might suppose he doesn't debate that because he knows it's a bad justification, or he doesn't actually want to risk concluding that God doesn't exist, while a more generous assumption would be that he merely recognises the subjective and personal nature of his true justification, so he tries to offer other justifications for what he believes to be true). But this renders it somewhat pointless for an atheist to debate him on those philosophical arguments, because for Craig, debating whether such an argument is sound would be independent of the important question of "does God exist"... although there may still be some utility in such a debate for the audience. I expect the same could be said for many apologists, but there may not be that many who would admit to it.
