There is no clean way to parse "More important, ..." strictly as written, but the intent is easy to understand.
A critical point is in the answer you reference:
...the adverbial phrase ‘more importantly’ modifies nothing in the sentence. What is wanted in constructions of this kind is ‘more important,’ an ellipsis of the phrase ‘what is more important.’
(Note that this is a subject with some disagreement, and the above quote expresses one opinion (of many) in the debate.)
If we accept that "More important, ..." is short for "What is more important is that..." then there is no issue at all: important modifies the fact expressed in the that clause.
Consider the following similar sentences:
Finding the treasure is important.
It is important that we find the treasure. (using an expletive it)
If we ask "What is important?" the answer is "that we find the treasure". (Or, "finding the treasure" in the first setence.)
You can also read "More important, ..." simply as:
This fact is more important: ...
"More importantly, ..." doesn't usually modify a verb in the main clause:
We lost the the treasure. More importantly, we lost our friends in the woods.
You didn't importantly lose your friends -- that doesn't make too much sense. Rather, you lost your friends, and that fact is more important than the fact about the treasure.
Rule of thumb: You can always omit that after the reason (i.e. the reason (that)). Put it another way, wherever you can say the reason that, you can omit that.
That vs. Why: You can use that (which is optional) in place of why but only in defining clauses, and make sure that you use reason not reasons (plural) if you want to use a that-clause. (See Cambridge Dictionary Online's note below.)
Question: [W]hat are the similarities or differences between the two sentences below?
- He didn't tell me the reason that he wore a polka-dot dress.
- He didn't tell me the reason he wore a polka-dot dress.
Answer: There is no difference in meaning whatsoever.
References†:
Macmillan lists these uses of reason, among others:
1 [countable] a fact, situation, or intention that explains why something happened, why someone did something, or why something is true
reason for: The police asked her the reason for her visit.
reason for doing something: Could you explain your reasons for choosing this particular course?
reason why: The reason why so many people caught the disease is still not clear.
reason (that): The reason these cars are so expensive is that they are largely built by hand.
[...]
Cambridge Dictionary Online lists reason why, reason that, and reason + to-infinitive.
Reason why We use reason why before a clause. We often omit why, especially in statements:
The reason (why) I didn’t contact you was that I was only in town for a few hours.
Reason that We use reason that before a clause. We often omit that, especially in statements. Reason that is less common and slightly more formal than reason why:
The reason (that) we need new guidelines is that the present ones are just not working.
We do not use reason in the plural with a that-clause:
There are several reasons why I don’t like the book.
Not: There are several reasons that...
Reason + to-infinitive We can use reason with a to-infinitive:
There’s no reason to be suspicious – everything, is perfectly normal.
Longman English Grammar (by L. G. Alexander) mentions reason why, reason for which, and reason that shortly.
1.38.3 Reason
defining:
That's the reason (for which) he dislikes me.
That's (the reason) why he dislikes me.
non-defining:
My success in business, (the reason) for which he dislikes me, has been due to hard work.
My success in business, the reason why he dislikes me, has been due to hard work. (The reason cannot be omitted before why.)
1.38.4 ('That') in place of 'when', 'where', 'why'
That is possible (but optional) in place of when, where and why but only in defining clauses:
[...]
That wasn't the reason (that) he lied to you ((That) can be replaced by why or for which)
†Even though you've asked specifically for the exclusion of reason why, I decide to keep all alternatives--reason why, reason that, reason for, reason for doing something, reason to do something--in the above quoted texts, because it could, I believe, make this question more useful for the future reader.
Best Answer
There is no problem with modifying prepositions, such as like or many others where a notion of "how much" makes sense.
Straight modifies toward, expressing "how much toward" (in this case, directly or straight toward).
Prepositions that express "targeting" (i.e. prepositions that identify an object of a phrase) obviously won't make sense with many modifiers, such as in most cases of for, to, from, etc.