What is the foundation of me experiencing the present moment being right now and not at some other point in time?

Under the premise of spacetime being a static and eternal thing, doesn‘t any line of thought trying to answer this question necessarily make any intuitive notions of identity and the passing of time illusionary?

Illusory in what sense? The underlying laws of our universe are time symmetric^[1], but the second law of thermodynamics means that entropy increases as you move away in time from a set low-entropy point (the big bang). This means that predicting bits at $t+1$ from bits at $t$ tends to be a much more difficult exercise than predicting bits at $t-1$ from bits at $t$. Large amounts of detailed information (entropy) about $t-1$ can, with some tricks, often be read off from $t$ with little computational effort. "Memory" is one way to do this. 

There are fewer ways to read off lots of information about $t+1$ cheaply from $t$. It is only possible in some very specific situations. You could, for example, look at a couch in a windowless room at time $t$, and commit to look at that exact couch from that exact position at that exact angle again one week later. This would let you pretty reliably infer a large batch of future visual bits. But such techniques do not tend to generalise well, you can't do this for arbitrary future visual information the way you can use "memory" to do so for a wide class of past visual information. Thermodynamics means the trick only works one way, for bits that are closer in time to the big bang. To do the same for future bits is theoretically possible, but it typically requires different techniques and a far, far larger compute investment.

For an observer operating under such physics, it is useful to conceptualise the world as consisting of a "past" that has "already happened", and is thus amendable to inference through techniques like memory, a "future" that has "yet to happen" and is more uncertain, and a kind of "present", where these regimes meet, close to which techniques for inference from both regimes tend to be most efficient, and where information can be processed directly, because physics is local in time as well as space. Thus memory from $t-ϵ$ with $ϵ<<1$ tends to be easier to protect from degradation than memory at $t-1$, computing bit predictions for $t+ϵ$ tends to cost way less than for $t+1$, and so on.

If you look at an intelligence inside such local physics, you will see that its internals at any point in time $t$ tend to be busy computing stuff about $t$, the "present moment" which the computation can locally operate on and affect, and which can often take in information from the "past", especially the "recent past" fairly easily, but has a harder time taking in information from the future, to the point that doing so usually involves totally different algorithms which feel totally different [LW · GW]. So it feels to the computation at $t$ that "it exists", "now".

I wouldn't really call this an illusion, except in the sense that "trees" are an illusion. A tree is fundamentally just some quantum field^[2] excitations from a particular class of excited states inside a finite 4D volume. But its medium scale, medium-range interactions with baryonic matter are often pretty well described by the human concept of "tree".

 Likewise, dividing time into a "future", which has "yet to happen", a "past", which "has happened", and a "present", which "is happening", is a leaky abstraction of the underlying laws about performing inference and decision computations in a physics with locality, the second law of thermodynamics, and a low-entropy state at some $t_0$ (big bang). It's not precise, but a good approximation under many circumstances. 

Imagine someone in the desert thinks they see an oasis. If it is actually a mirage, I'd say it makes sense to call the oasis an illusion. If it is an actual oasis, I don't think the moniker illusion is apt just because oases are really an imperfect abstraction of particular quantum field configurations.

1. ^{^}

   Well, actually CPT symmetric, but the distinction doesn't matter for the intuition here.

2. ^{^}

   Or, if you don't believe in asymptotically safe quantum gravity, it might not really be quantum fields either. Substitute whatever your favoured guess for the true fundamental physical theory is.

State of mind that describes the experience of 11:28 AM exists in my mind at 11:28 AM, that's what the brain does, it prepares this state of mind using its structure, preceding states of mind, and observations. At 11:30 AM, the state of mind is different, and describes the updated experience of 11:30 AM.

A device that instantiates my state of mind of 11:28 AM at 10:28 AM would result in a me that experiences 11:28 AM at 10:28 AM. With this device, there are two mes of 11:28 AM that exist at different times, both at 10:28 AM and at 11:28 AM. If I'm aware of the device, the me of 11:28 AM knows that now is both 10:28 AM and 11:28 AM. It's a foothold for controlling the past [LW · GW].

It's a hard job, but someone's gotta do it.

I suspect it is creation of memories. You don't experience time when you're not creating memories, and they're some kind of very subtle object that lasts from one moment to (at least) the next so they leave a very subtle trace in causality, and the input that goes into them is correlated in time, because it is (some small selection from) the perceptions and representations you had simultaneously when you formed the memory.

I even believe you experience a present moment particularly intensely when you're creating a long-term memory - I use this to consciously choose to create long-term memories, and it subjectively seems to work.

You have a subjective distinction between “now” and all earlier times.  Evolution made you that way.  At any given moment, much of the maximally informative information about what do now will have arrived in the past fraction of a second.  Information from a minute ago, or last week, may be useful, but our mind makes a distinction between the freshest information and the old stuff.  The freshest information is called “now”.  We feel like we’re deciding things and doing things “now” because the upcoming fraction of a second is the time over which we have the most influence.

”Now” is not a fundamental fact about the universe, it’s a useful abstraction in an algorithm that has to steer a body through a difficult world.  Its subjective salience is what it feels like to be that algorithm.

"Now" is the time at which you can make interventions. Subjective experience lines up with that because it can't be casually compatible with being in the future, and it maximizes the info available to make the decision with. Or rather, approximately maximizes subject to processing constraints: things get weird if you start really trying to ask whether "now" is "now" or "100 ms ago".

That's sort of an answer that seems like it depends on a concept of free will, though. To which my personal favorite response is... how good is your understanding of counterfactuals? Have you written a program that tries to play a two-player game, like checkers or go? If you have, you'll discover that your program is completely deterministic, yet has concepts like "now" and "if I choose X instead of Y" and they all just work.

Build an intuitive understanding of how that program works, and how it has both a self-model and understanding of counterfactuals while being deterministic in a very limited domain, and you'll be well under way to dissolving this confusion. (Or at least, I've spent a bunch of hours on such programs and I find the analogy super useful; YMMV and I'm probably typical-minding too much here.)

I don't have precise answer to your question, but have some question which can prehaps be useful in answering it. 

Namely: what about space? I mean, you talk here about time, something along the lines of "We can imagine our Universe as a solid eternal block of spacetime, so why do I experiencing present moment instead of all moments at once?" But what about "We can imagine our universe as a solid eternal block of spacetime, so why I am experiencing 'here' locality instead of all places at once?". I think these are very similar questions. 

We can go even further and ask "We can imagine Multiverse of self-consistent mathematical objects (among which is, of course, our Universe) as one eternal blob, so why do we experience present, local moment instead of everything existent?". 

It seems, the key to confusion here is that we imagining ourself as an outside observer of arbitrary high level and then we ask "why our experience difers from experience of such an observer". But this is exactly because we are not this observer. What we are is a different question, of course. I personally havn't yet found a satisfactory answer to the hard problem of consciousness, but it seems this specific in experience (space/time locality) tells us about something about which consciousness is certainly not.

Time, change, and continuity through change, are all real. A life is not a constellation of static experiences in eternity, disconnected but imagining that they are connected; it is the history of the changes experienced by one awareness as it transforms. The phenomenological study of experienced time will tell you about its nature, not the current guesses of physicists and neuroscientists. 

I pretty much agree with your hypothesis. Each 'moment' of conscious experience is a distinct, unique -- something. Our subjective stream of consciousness is simply the most likely path through all possible spacetime states that lead to the 'present' -- sort of like a Feynman sum-of-paths integral.

Not sure how to fit quantum mechanics in there...