Let me try to reply to this in sequence, partly based on what's given in the question and partly on what was relayed in some comments.
It seems to me there are simply too many it's in use today.
The word is an important element of English, and I don't agree that there are too many of them. It seem obvious that you don't like the word for some reason, but I think you'd be in a minority when it comes to actually advocating against it—or wanting to change our language to remove it.
It occurs to me . . .
In my opinion . . .
You asked why have the first sentence form instead of the second. In my view, neither sentence is an "improvement" over the other. There is a subtle difference of meaning (I interpret the first to be about you becoming aware of something and the second to be about you already having a belief), but nothing really noteworthy. The choice between the two is simply a matter of personal preference.
But compare the full version of the original to something with even greater parsing:
It occurs to me there is a much simpler solution to the problem.
There is a simpler solution.
Now that does make a difference. It's far simpler and more direct. The meaning has been changed, but I don't believe greatly enough to lose its essential quality. (Especially if context has already been established for the sentence.)
Here, it's not about replacing it with a non-it word, but actually using the fewest number of words to make the same basic point. It just so happens that in removing half the words, it was one of them.
For example, why couldn't the following sentence lose the it without damaging the sentence and its meaning?
It is a travesty the way President Trump uses the English language.
Try:
President Trump's use of English is a travesty.
That's shorter and more direct. It has been removed, but I didn't rewrite the sentence with that intention specifically.
Let's look at some more simplifications:
It is hard to conceive how best to solve the immigration problem we have in America.
America's immigration problem is hard to solve.
It appeals to me the way Joe Blow expresses himself.
I like how Joe Blow expresses himself.
It's a matter of taste when it comes to evaluating how appropriately and skillfully food is seasoned.
Skillful food seasoning is a matter of taste.
All of these rewrites remove the word it. But it isn't the use of it specifically that was the problem. It was the construction of the sentences that all began with it. Which isn't to say that beginning a sentence with it is always bad:
It's raining.
It was a poor attempt.
Here, I can think of no way of making the sentences shorter or clearer by removing the word it. In fact, if they were rephrased to remove that specific word, I think the sentences would be slightly awkward:
There is rain.
A poor attempt was made.
Let's consider a rephrasing example you gave in a comment (which I have simplified slightly):
It is not difficult to understand Strauss's about-face.
To understand Strauss's about-face is not difficult.
I find the rephrased version more difficult to parse. Why?
Verb phrase: (It is not difficult to understand)
Noun: (Strauss's about-face)
Verb phrase, part 1: (To understand . . .)
Noun: (Strauss's about-face)
Verb phrase, part 2: ( . . . is not difficult.)
In the rephrased version, the flow of reading the verb phrase is interrupted by the noun. This is an annoyance, and makes the understanding of the complete sentence reliant on an interrupted thought, since full parsing can only happen in "hindsight."
Let me suggest a different rephrasing:
It is not difficult to understand Strauss's about-face.
Steve's about-face is easily understood.
I find this much better. There is no "interruption" and everything is easily parsed without it.
Where does this leave me?
I actually think that, in analysis, you are partially correct in saying that if you rephrase certain sentences to remove it that it results in a simpler, more easily understood sentence.
My initial opposition to this suggestion was coloured by the fact that I found your rephrased versions more difficult to read. Not because they were missing it but because the rephrasing involved "mental interruptions."
However, I don't think that it's actually it itself that you object to but the fact that it makes vague some sentences that have been too-quickly constructed with it. Your objection isn't to it per se, but to a lack of plain language in general.