How does an ordinary person adopt Right View concerning the the hardships of the Buddha's earthly life?

Well first putthujjanas won't know anything about the dhamma until they become at least sotapanna. Until this, all puthujjanas have is believes and doubt. It's completely normal. All the puthujjanas can do is cultivate what the buddha recommend to be cultivated, that is the 8 fold path. This is pretty hard for anybody. It's worse when there is no buddha.

Since the big claim of the buddha was that craving causes dukkha, you test this and can keep a diary for a few weeks where

• you note when you notice a craving
• you note the object of your craving
• you note when you do not crave some objects and in this case if you suffer from not-craving those objects
• you note when you suffer
• you note if you think that you suffering can be linked to some craving for something (ideas or just sense objects) and if you see them as part your self
• you note what happens if you try to force to view as not-self all the stuff you so far view as self

OP: Here is the problem: we know that the Buddha suffered hardship in this world, including sickness and requiring a physician at times. We know that he had enemies who wished him harm. How then is it possible to believe that the Buddha was the happiest or most fortunate person ever to exist? Surely, out of all of history, with billions of people, there must have been somebody with fewer sicknesses and fewer enemies!

This is exactly the point!

Freedom from suffering is really freedom from mental suffering.

The body can suffer from disease, family members can die, friends can forsake you, wealth can dwindle, things you desire can be taken away from you or denied from you, but in all cases, you have a choice whether to suffer or not, because suffering is of the mind.

The Buddha endured his physical pains but he did not suffer from them.

The Buddha taught that all conditioned things are impermanent and suffering, and so we should not peg our happiness to them.

From SN 36.6:

“Bhikkhus, when the uninstructed worldling is being contacted by a painful feeling, he sorrows, grieves, and laments; he weeps beating his breast and becomes distraught. He feels two feelings—a bodily one and a mental one. Suppose they were to strike a man with a dart, and then they would strike him immediately afterwards with a second dart, so that the man would feel a feeling caused by two darts. So too, when the uninstructed worldling is being contacted by a painful feeling … he feels two feelings—a bodily one and a mental one. ...

“Bhikkhus, when the instructed noble disciple is contacted by a painful feeling, he does not sorrow, grieve, or lament; he does not weep beating his breast and become distraught. He feels one feeling—a bodily one, not a mental one. Suppose they were to strike a man with a dart, but they would not strike him immediately afterwards with a second dart, so that the man would feel a feeling caused by one dart only. So too, when the instructed noble disciple is contacted by a painful feeling … he feels one feeling—a bodily one, not a mental one.

In the essay "To Suffer Is an Active Verb", Ven. Thanissaro wrote:

When we say that we suffer, we usually think that we’re on the passive receiving end of the suffering. It’s something imposed on us, something to which we have to submit. In some of our more mature moments, we realize that there are times when we’re adding to our own suffering, but we tend to see this more clearly in other people than we see it in ourselves. This is one of the reasons why, when we come to the four noble truths, the Buddha’s analysis of suffering, we have to take them on faith—because, in his analysis, to suffer is an active verb. It’s something we’re doing actively. It’s a choice we make. It’s a choice we make badly, out of ignorance. The suffering is in the activity of clinging.

WHY does one believe, that he was totally free from misery or mental anguish?

Perhaps you can see the "Noble Truths" in your own life -- suffering, cause of suffering, cessation of suffering.

Given those are true, based on your own personal experience, perhaps it's not too hard, perhaps it's plausible, to imagine that someone (the Buddha) who trained himself full-time -- admittedly someone extraordinary -- could realise those truths "completely" (and then teach them well).

Should one believe that Buddhism can bring earthly blessings, or not?

It depends on your definition of "blessing".

If you believe that happiness and sorrow follow from the mind, then a "tamed" mind is a blessing.

That's a primary message of the Dhammapada, in chapter 3 and the opening of chapter 1:

1. Mind precedes all mental states. Mind is their chief; they are all mind-wrought. If with an impure mind a person speaks or acts suffering follows him like the wheel that follows the foot of the ox.

2. Mind precedes all mental states. Mind is their chief; they are all mind-wrought. If with a pure mind a person speaks or acts happiness follows him like his never-departing shadow.

And there's a Mangala Sutta which lists various "blessings".

Buddhism isn't magic though, causing blessings to appear -- see for example SN 42.6 -- but it points towards what's achievable ...

The Dhamma is well declared by the Bhagavā:
visible here and now, immediate,
inviting to come and see, effective,
to be individually ascertained by the wise.

... and towards what's "right" to value (or to view), what's right to intend, what's right to practice, etc.

And if not, then how can a human being who suffers from a desire for/lack of earthly blessings form the motivation to follow the dharma whole-heartedly?

A motivation might be to gain liberation from suffering -- and an understanding of the path which leads to the end of suffering -- and "faith" (or personal experience) that the path is effective.

In the simplest terms, why would an ordinary human being with ordinary struggles want to live the life of the Buddha?

Yes and no.

• The life of the Buddha included renunciation, living as a monk. In my experience (I'm living in a non-Buddhist society) that's far from what most ordinary humans tend to aspire to as an ambition or even an option.
• The life of the Buddha was also kind in various ways, moderate, wise, a good friend, harmless moral and generous. That might be within the scope of ordinary human ambition -- perhaps what children of "good families" learn as values from their parents or other teachers, and any good friends they might have met.

Is it even possible for an ordinary human being with a difficult life to want to be like the Buddha?

I'm not sure where or how the path starts.

Some of the fundamentals, the bases, include:

• Hearing the Dhamma (to form an understanding of what the Buddha is like, what it might mean to "be like" the Buddha)
• Wanting to not "suffer", to end suffering, to not cause further suffering (to oneself or others)
• Morality and generosity