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KS - I find the image of the fire helpful, as well as important to consider in context as fire imagery was a prominent image for the Brahmans.

Thanissaro Bhikkhu has an excellent article on Nibbana that I think is relevant
According to the ancient Brahmans, when a fire was extinguished it went into a state of latency. Rather than ceasing to exist, it became dormant and in that state — unbound from any particular fuel — it became diffused throughout the cosmos. When the Buddha used the image to explain nibbana to the Indian Brahmans of his day, he bypassed the question of whether an extinguished fire continues to exist or not, and focused instead on the impossibility of defining a fire that doesn't burn: thus his statement that the person who has gone totally "out" can't be described.

When the fuel goes out for the candle or for what I call "me" it is not relevant to talk about where either of us have gone.
To answer this question, I would certainly not posit a latent state for the fire nor would I posit a latent state for the "me" that goes out - that is the antarābhava of the Sarvāstivāda.

From here, I simply try to keep in mind what the purpose of this teaching is - liberation from suffering and specifically from the suffering arising from the grasping at the false sense of self, which you've so clearly articulated.

When I understand this grasping at self as the root of attachment and aversion, it begins to give rise to some sense of detachment that then lessens my attachment and aversion.

The light for me is that concerning myself with where the fire goes when the candle burns out is not conducive to liberation, so I can let it go.

As Thanissarro Bhikkhu ends the article I've referenced above,
...the next time you watch a fire going out, see it not as a case of annihilation, but as a lesson in how freedom is to be found in letting go.

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The analogy of a lighted candle passing on its light to another candle does not really work for me because in that case, we are still talking about a physical phenomena: the present and future bodies (the candles) being both present when consciousness (the fire) is being passed on.
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First of all, human rebirth is NOT taken for granted...several preconditions shall be in place. Second, alternatives to the rebirth theory such as eternalism or annihilationism are explicitly discarded by Gautama Buddha in the pursuit of the middle path.

"Two Views on Rebirth
The eternalist doctrine holds that this present, existing person will survive after death, will
continue, will go on more or less intact, to take up residence here or there in the future.
The annihilationist doctrine holds that this present, existing person will be annihilated at death
– destroyed, wiped out, obliterated – and there will be no future existence of any kind.
When the Buddha is asked questions about death and rebirth, he does not side either with
eternalism or with annihilationism. Instead, he expounds the deep and subtle doctrine of
dependent co-Arising.
All living beings are dependently arisen; they exist and change as patterns of conditions
dependent upon other conditions. Life does indeed go on from one existence to another, but it
is not identical life – it is just the natural consequence or outcome of what has gone before
-...-
It is quite evident that
the main doctrine of Buddhism, the theory of Dependent Origination, clearly rejects the
eternalism of Brāhmaṇās as well as the nihilism of Śramanās.20 The religious practice, the
Middle path formulated by the Buddha, also goes against the practice of self indulgence
(Kāmaskhallikānuyoga) of the materialistic school and the practice of self mortification
(Attakilamathānuyoga) of the Jaina school
"
You will NEVER get a scientific demonstration of these theories...nobody can put any on a scale to show you the exact weight down to the third decimal point that is as acceptable in Alaska as it is in Bangladesh.
Doubts (much as greed, hatred, delusion, sloth &torpor) are to be overcome, but Gautama Buddha or mahasattva Ksitigarbha cannot take you beyond the obstacle. Buddha died at age 80 because he consciously chose to eat adulterated food to reach parinirvana...and his hand was empty...
You may take the teachings or leave them...Buddha won't threaten anyone with pains of eternal death...as if life weren't enough of a threat as it is...

I believe your difficulty relates directly to "anatta" (non-self). You seem to write with the belief that your 'self' exists in itself (or can be "stored" independently).

I tend to use two analogies when questionned on rebirth, maybe they'll help you. Let me know...

First, continuity both is and is not. Consciousness is like a river. Now, from one second to the next, the water particules move, whirlpools appear and move and disappear, waves appear, change, move, disappear... so from one moment to the next, this is not the same river, the river has no self... but (most likely, unless a dam appears upstream, or some other massive change in conditions) there still is a river where there was one a moment ago: conditions constantly change but, if they don't change too drastically, what was resulting in a river a moment ago is likely to result in a river now. Consciousness through rebirth is the same: the world in which we live will be little affected by someone dying, and if the conditions a moment earlier were conducive to the appearance of consciousness (via the classic chain contact-sensation-feeling-perception-consciousness...), then the conditions having little changed, it still is conducive to make consciousness appear.

The second analogy uses co-dependence... You condition your environment, the environment conditions you. As such, by your acts, you shape the world around you (it shapes you too). A key point is that how you shape your environment says something about you.
Can you imagine for a second that you may shout very loudly some information and a few seconds later, the echo would come back to you? Now imagine you die after the shout and before the echo, and you're reborn immediately. The echo arrives and gives information to the new being, right? There is continuity (the very same idea even!) but it is via the environment (the echo), rather than _directly_ from one body to the next body... The chain is more like mind-environment-mind, rather than mind-mind, but there is a chain and a continuity from the "old" mind to the "reborn" mind. So you do not need the two minds to be simultaneously present.
Now, as I said, you shape the environment by your acts, and the environment thus says something about you. So there's no need for a shout, all your acts are 'shouts'... and there's no need for an echo, all your perceptions are perceptions of an environment you shaped, everything you perceive is an echo...
If you want to relate this to the candles (although the analogy is just an analogy, you shouldn't push it to the extreme), you might want to realise that the flame can pass from one candle to the next without the candles touching each other... continuity can be created without contact, thanks to a medium (be it fire, or the environment, the world you contributed to shape...). And it depends on the school, but my understanding is that e.g. in Theravada, it may be considered that rebirth is immediate and is also conditionned (it requires a fertile couple) so the two candles _are_ present: the dying person, and the creation of the first cell of the new person, the two are present but they don't need to touch one another directly. The environment creates the connection, just like the fire does.

Kind regards
Denis