Your "correction" is actually awkward and makes very little sense. I'll try to explain the grammar behind the original text. You have two parts of the sentence. I've written them neutrally, with the proper word-order: "These time-based technologies have become so integral to day-to-day existence..." "...that our dependency on them is recognized only when they fail to work." However, there are two problems. 1) You want to emphasise how integral they have become, and 2) you want to emphasise the second part of the sentence above the first part. Using inversion will do that nicely. Therefore, "These time-based technologies have become so integral" becomes "So integral have these time-based technologies become..." It seems to follow a question-order, but is not a question. There's a larger grammatical rule at work here. Look up "inversion" (it's advanced grammar) and try some exercises.

The first option is correct, but sounds a bit awkward. The second option is incorrect because of the "have become TO day-to-day" portion within "So these integral time-based technologies have became to day-to-day~..." I would suggest something closer to "These time-based technologies have become so integral in our day-to-day existence that dependency on them is recognized only when they fail to work." Your current sentence is correct though. It's up to you if you would change it.

The change that you wish to make is not correct.