Collatz question

For the Collatz Conjecture, why can't one rationalize the solution by omission of the possibility of any numbers that break the pattern of collapsing to 1?

My train of thought is thusly: All real integers greater than 0 must be either odd or even. Taking a random number from any number within the set of real integers, it has a 50% chance to be even, 50% chance to be odd. Numbers ending in 0, 2, 4, 6, or 8 are even, and numbers ending in 1, 3, 5, 7, or 9 are odd. Of these, we look at evens and odds respectively, treating each part of the Conjecture, 3n+1 and n/2 sort of separately.

Taking even numbers, roughly 50% of even numbers will yield an even number, and 50% will yield an odd number. Taking 2, 4, 6, and 8, a number ending in those digits will end with a 1, 2, 3, and 4, respectively. Looking at 0, a number can either be odd or even depending on the the digit to the 10^n+1 place where n is the place of the digit in question. If it is odd, then the 0 will yield odd, if even, it will yield even (30 -> 15, 40 -> 20). The number can be followed by multiple zeroes so in that instance, the 0 becomes the subdivision of new probabilities, 50% odd, 50% even, so on and so forth.

Taking into account only odd numbers: Putting an odd number in the "Collatz function" will necessarily yield an even number, because an odd multiplied by an odd will always yield an odd number, which added to 1 creates an even number. If every odd must inherently become even, then after once cycle of the fuction applied to any random number within the set of real integers, from 50% of integers being odd, and 50% being even, you are left with 75% being even to only 25% being odd. You can continue this cycle to create 87.5% even to 12.5% odd, and so on. Since the limit of the powers approaches 100% of numbers generating even numbers, should it not be possible to assume that all numbers must reduce by powers of 2 but cannot become lower than 1, then the series must converge at 1?

I know that this isn't a mathematical proof, strictly speaking. I was just curious about the concept of the conjecture and the fact that it seemed to not be able to be definitively solved or disproved. I'd appreciate enlightenment or other resources for my curiosity. Out of what I have seen on the subject matter, no one has approached it this way (and probably for a good reason, like having a deeper understanding about math). I'm more curious as to why this method isn't considered to be defining, persuasive, or a good enough rationalization.