I already stated that I'm not a Buddhist and I don't believe in the Buddhist teaching of rebirth. I am very interested in Buddhism, however.

it doesn't follow that you have practised since beginningless times and you have already practised with diligence infinite times and you somehow always failed. — boundless

If a cycle of rebirth and death is beginingless then there will always be a previous cause or rebirth and this would go back infinitely. If there’s no beginning then there’s no end.

This isn't necessarily the case. Traditional buddhists would reply that the ultimate cause of the cycle is ignorance. If ignorance is removed, samsara stops. If ignorance is never removed, the cycle will go on forever.

BTW, this problem was one of the reason why I ultimately ceased to try to become a Buddhist. If the cycle is beginningless, then the very existence of the 'cycle' is unintelligible. As youb remarked, each instance of rebirth is intelligible in principle, it is a regulate phenomenon. It would be weird if the very existence of the cycle is an unintelligible 'brute fact'.

If, however, the cycle began, this means that if other traditional Buddhist claims are true it must end:

“Whatever is subject to origination is all subject to cessation.” — SN 56.11, bhikkhu Bodhi translation

However the traditional Buddhist view is that it doesn't necessarily end. Rather it ends if ignorance is removed.

If the cycle is beginningless, then the very existence of the 'cycle' is unintelligible. — boundless

'Saṃsāra has no beginning, but it has an end. Nirvāṇa has a beginning, but it has no end' ~ Buddhist Aphorism (quoted on Dharmawheel.)

This isn't necessarily the case. Traditional buddhists would reply that the ultimate cause of the cycle is ignorance. If ignorance is removed, samsara stops. If ignorance is never removed, the cycle will go on forever. — boundless

But we’re not looking forward, we’re looking infinitely backwards, and in the past ignorance has necessarily never been removed because we are here in ignorance.

This ventures into some concepts more native to some schools of Hinduism, with the veil being the "veil of Maya".

Yes, my position is more on the Hinduism side of the issue (via Theosophy)

The problem with assuming defaults, innate essences (such as "all beings have Buddha nature") is that they bog one down.

Assumed for the purpose of discursive discussion.

If you have Buddha nature, then why are you here, suffering, instead of being happy and enlightened?

One is going through a process, there may be many other things going on (behind the veil), or of which we are a small part. Which entail what is going on here. One of the first things that occur to us as individuals as a young child is the realisation of our individuality and therefore questions arise about our circumstances, what is going on here, where is this, why am I here? I remember this realisation in my life, I must have been about 3yrs old. These questions have not been answered, even though I have searched long and hard for an answer. As such there cannot be an answer for your question, because the circumstances relating to it have not been established.

If you suffer now, despite having/being Buddha nature, and later become enlightened, then where's the guarantee that you won't lose your enligtenment and suffer again?

Again this can’t be answered, as above. However, presumably, one would have sufficient agency to prevent the onset of suffering. Although I would suggest that there is likely an exalted state equivalent to suffering within that exalted realm. On the cosmic scale, there may be imperfect gods, or greater processes beyond our understanding going on.

If you are now covered by the veil of Maya, how can you possibly trust your choice of spiritual guidance?

Through humility and faith. This would necessarily require living a relatively simple and stress free life.

Thus assuming some kind of innate natrure, an essence, implies, among other things, that you are ultimately helpless against that veil of Maya, helpless against suffering.

I’m not quite sure where the implication lies here. But never the less, when one thinks about our circumstances, we are as individuals helpless. We rely entirely on our community for almost everything. When it comes to salvation, we might think that we personally somehow achieve something, but what is more likely is that circumstances bestow it upon us. As we are playing a small part in a greater process. A process which given we are talking about “supernatural” states like nirvana, will likely entail transcendent realities beyond our comprehension.

It's how the outlook of innate nature is demoralizing, unless, of course, one has a grand enough ego to compensate for it.

Or perhaps it is an acceptance in humility of a reality. Presumably, by this point one would have deflated and reconciled one’s ego.

I actually find both rebirth and reincarnation entirely plausible.

I also find the Hindu explanation plausible according to which Vishnu/Krishna incarnates himself as a buddha/the Buddha.
Having studied a bit of both Buddhism and Hindusim, I find there is a peculiar fit between the two.

Likewise.

As @baker remarked, the idea is quite explicit in some strands of Mahayana with the concept of 'Buddha nature'. However, it can be said that it is implied by the fact that the Buddhist practice is seen as a way to purify the mind, i.e. removing all the 'impurities'. So, rather than a transformation into something 'alien', the Buddhist path actually seems to have been presented as a way to bring the mind-stream to its 'purity'.
This idea is IMO recurrent in ancient religious and philosophical traditions. You can find analogous idea in Christianity, for instance, when sins are depicted as an impurity or an illness that 'stain' the purity (yes, there is original sin but as you probably know the interpretation of that concept wasn't the same among all Christian traditions... and, anyway, there is the idea that all God's creations are originally good and, therefore, evil is a corruption that came about later).

Yes, so my intuition is actually an acceptance (or realisation) of a deeper understanding underlying these religions. That they are playing a role in a process of purification of the self. That the self is not required, to go anywhere, to do anything, achieve anything in reconciling (becoming liberated from) their incarnation. But rather to relinquish, to lay down the trappings of our incarnate selves.