Surely phrases like ר׳ יהודה אומר weren’t around from back then. I understand the concept to mean that certain halachos were around from Moshe’s time and were passed down, generation to generation, until Rebbe, who then wrote down the Mishnayos. In the interests of keeping things brief, he would write certain halachos explicit, while others he would leave implicit (תנא __ אין מכלל ד__ לא). Some he still left in Braisos even after all that, hence R’ Chiya’s and R’ Oshaya’s job of compiling the Tosefta. Sometimes Rebbe would couch two seemingly disparate halachos together, such as reading the Shema when the Kohanim Tevulei Yom could eat Terumah.
Now, that begs the question still: even if the precise words might have changed over time, how did they ensure that the Halacha wasn’t garbled? This is one of the most common questions on the concept of Mesorah. Have you ever played telephone? The phrase that’s put into the chain is never the same one that the last guy hears.
Well, that’s not quite an accurate metaphor for how it actually went down. Let’s play telephone again, but after the second guy says his piece, the first guy gets a chance to say, “Hold on a second, that’s not right.”
The following is from Eruvin 54b:
ת"ר כיצד סדר משנה משה למד מפי הגבורה נכנס אהרן ושנה לו משה פירקו נסתלק אהרן וישב לשמאל משה נכנסו בניו ושנה להן משה פירקן נסתלקו בניו אלעזר ישב לימין משה ואיתמר לשמאל אהרן רבי יהודה אומר לעולם אהרן לימין משה חוזר נכנסו זקנים ושנה להן משה פירקן נסתלקו זקנים נכנסו כל העם ושנה להן משה פירקן נמצאו ביד אהרן ארבעה ביד בניו שלשה וביד הזקנים שנים וביד כל העם אחד
The Gemara continues to discuss methods of Torah study. The Sages taught the following baraita: What was the order of teaching the Oral Law? How was the Oral Law first taught? Moses learned directly from the mouth of the Almighty. Aaron entered and sat before him, and Moses taught him his lesson as he had learned it from God. Aaron moved aside and sat to the left of Moses. Aaron’s sons entered, and Moses taught them their lesson while Aaron listened. Aaron’s sons moved aside; Elazar sat to the right of Moses and Itamar sat to the left of Aaron. Rabbi Yehuda disagreed with the first tanna with regard to the seating arrangements and said: Actually, Aaron would return to sit to the right of Moses. The elders entered and Moses taught them their lesson. The elders moved aside, and the entire nation entered and Moses taught them their lesson. Therefore, Aaron had heard the lesson four times, his sons heard it three times, the elders heard it twice, and the entire nation heard it once.
נסתלק משה ושנה להן אהרן פירקו נסתלק אהרן שנו להן בניו פירקן נסתלקו בניו שנו להן זקנים פירקן נמצא ביד הכל ארבעה
Moses then departed to his tent, and Aaron taught the others his lesson as he had learned it from Moses. Aaron then departed and his sons taught the others their lesson. His sons then departed and the elders taught the rest of the people their lesson. Hence everyone, Aaron, his sons, the elders and all the people, heard the lesson taught by God four times.
Now everyone knows the same halachos as everyone else. From this point on, the Oral Law and the Written Law are no different - if someone tries to change the Halacha, or if he does so unintentionally, he has millions of people contradicting him.
It’s not a perfect system. Things will still change over time. But it’s the best system possible.
What other options are there? Each person make it up as they go? You’ll get several different “authentic” Judaisms very quickly. Passing down the Oral Law through a centralized body which everyone obeys is the best way possible. You’re correct that it’s not 100% foolproof, but that’s only going to happen if prophecy were to have continued to the present day and beyond.