"can you explain the logic you think is being failed?"
You can not have an infinite series of something, followed by something else. One definition of "infinite" is that it never ends, which is also why it isn't a number (a number is intrinsically something that would have a definite end). So to have "an infinite series of something, followed by a 1" is to have "a series of something that never ends, followed by a 1 after the end". I can type those words without crashing English, but it has no meaning.
"The alternative (that 1/3 is exactly 0.3333...) seems equally arbitrary to me."
No. I'm not even sure what else to say. If you need convincing, start the long division on paper and keep going until you're convinced.
No, seriously, keep doing it until you're convinced. Right now. Don't reply with a counterargument until you've done that. There's nothing arbitrary about it; to prove that there is, you need to show some point where you came to a choice and you chose to add the next 3, rather than some other hypothetical alternative.
"have come up with a fully consistent, alternative way of defining equality when involving infinite fractions that is no worse than the conventional definition."
Well, no worse, other than also defining an internally-contradictory definition of "infinity" for the sole purpose of winning an internet argument. Other than that, no worse, no.
Recall that introducing one contradictory premise into a logical system allows you to prove any statement. If you are not bothered by a "small" contradiction to prove a dubious point, you don't understand math. Thus, making up definitions to prove a point must be analyzed in the full context of a mathematical system, not just analyzed on some other abstract measurement system (that doesn't matter). Breaking infinity to win an argument doesn't work.
...
Most people go about proving this the wrong way. Proving something about the string "1" vs. ".999..." is wrong. In order to show the two are distinct entities, you need to show a situation in which they behave differently. No such situation exists. Until you do that, "1" = ".9999..." is no more surprising that "2/2" = "1". Or (ahem) "9/9" = "1". Numbers are what they do, they have no existence beyond what they do.
On that note, when teaching I think the point that "=" means "is bidirectionally fully substitutable by" is not taught properly. "2 + 2 = [underline]" is a flawed question; oh, we all know what it means, but "2 + 2 = 2 + 2" is a perfectly correct answer to the question as written. We ought to define a separate operator for what we really mean, "simplification", so we can do something like put "2 + 2 => [underline]" on a test, and save the equality operator for actual equality.
That is also why, for instance, the definition of the square root function must include the stipulation that it takes the positive root, because otherwise "sqrt(4) = 2" is not a true statement; if sqrt(4) means "both" roots somehow it is not fully substitutable with 2. (You can't say that both "sqrt(4) = 2" and "sqrt(4) = -2", either, because then "2 = sqrt(4) = -2". Oops.)
That's the background for my statement that .9999... = 1. It means that there's no way to tease the two apart with any (valid!) mathematical operation. Here's another way to say it: On the real number line, if you have real number A and real number B, and there are no points between A and B, then A and B must be equal. To show they are two different real numbers, you must show a number C between 1 and .9999.... You can not do this... well, again, you can't do this validly.
(Also, per RiderOfGiraffe's point, yes, there's some simplification here, but I think it's fair to run on the theory that intrinsic to this argument is that we're on the real number line. People who know enough to talk about transfinite ordinals don't usually get into these arguments; by then, they've learned the secret of math lies in the definitions chosen and understand the ultimately contingent nature of any answer. :) )