One in a thousand and Yehuda ben Teima
In Koheles 7:28, Shlomo writes "I have found one man in one thousand, (but a woman from all of these I have not found)." Rashi quotes the famous ma'amar Chazal (Vayikra Rabbah 2:1) that says, "The way of the world is that 1000 enter for mikra (Tanach). From these, only 100 go forth and succeed to be worthy of mishnah. Of these 100, only 10 go forth for gemara, and of those 10 who enter mishnah, only one goes forth for hora'ah." It seems to me to be saying that 90% of Jews should limit their learning to only Tanach, without touching the Torah she-b'al peh, the 9% who have succeeded (I suppose this means achieving mastery of the texts) should move on to the basic text of the mishnayos, and only 1% should move beyond the basic mishnayos to the discussions of the amora'im and the like. Practical halacha would likely not be included in this proscription against learning advanced materials, as such knowledge is necessary in order to properly perform the mitzvos and avoid aveiros, but the vast majority of learning Jews would be entirely ignorant of most of the yeshivish masachos, which would apparently not be a bad thing.
Perhaps one can tie this ma'amar Chazal into the statement of Yehuda ben Teima in the 5th perek of Avos, in addition to the statement of Chazal, derived from the Levi'im, that a student who does not meet with success after 5 years will never find success: A child has between the ages of 5-10 to master mikra. The 10% who do have between 10-15 to master mishnayos. The 10% from that group who do have between 15-20 to master gemara. Beyond that, ben Teima says that a person enters the phase of life "lirdof", to pursue, which is often explained as referring to pursuing a livelihood, as by this point, he should have either been filtered out of the yeshiva system and diverted into the working world, or he should have reached the level of basic hora'ah - but even then, being that one is not allowed to make a parnassah off of Torah, he would have to get a job anyway, like almost all of the Tannaim and Amoraim did. This is contrary to the common practice nowadays, where everyone is pushed on to the more advanced areas of learning, which often leads to a situation where people possess towers of Torah she-b'al peh knowledge built on stilts.
The other way to understand ben Teima would be in dissent to the other two ma'amrei Chazal, saying that *everyone* moves on to Mishnah and Gemara, despite the fact that they may not have acquired the requisite foundation to enable them to learn the more advanced materials properly. Perhaps one can defend the modern practice by drawing analogies between it and this reading of ben Teima, but the previous understanding of ben Teima seems to me to be more logical.