I imagine a lot of it will come down to the judgement of the parent and the nature of the child. No one methodology is more superior, it is about חֲנֹ֣ךְ לַ֭נַּעַר עַל־פִּ֣י דַרְכּ֑וֹ - Educating the child according to his (individualised) way (see Mishlei 22:6)
The education of our children is a very precious investment and is something that must be placed at the forefront of our priorities. It is about finding the balance that works best for the child. This idea is perhaps best illustrated with the following story related about Rabbi Elchonon Wasserman הי"ד. R’ Elchonon was once in London to collect funds for his Yeshiva in Baranovitch. Before leaving, he was asked by his host to bless his son. R’ Elchonon turned to the man and said, “When the Chofetz Chaim was asked by people to bless their sons to be talmidei chachomim he would tell them that you have to be willing to sacrifice. A parent must be willing to give away the pillow under his head to pay for his child’s tuition.” R’ Elchonon then proceeded to learn with the boy the piece of gemara he was currently learning. Seeing that he did not fully understand that particular sugya, R’ Elchonon took time to teach it to the boy until he understood it fully. R’ Elchonon then turned to the host and exclaimed; “Now this is an accomplishment”.1 We see from this powerful anecdote that the chinuch that our children receive is crucial. Whether it is attending a school vs homeschooling, the parents have to invest as their child's future is at stake. I don't think one option is better than the other, it is about finding a system that works best for the children. Some kids thrive if they have individual, personalised attention, others flourish if part of a classroom along with their peers.
To begin to understand the schooling system, one needs to first understand why it serves the purpose it does.
The Gemara in Bava Basra 21a famously outlines the progression of our educational system and praises Yehoshua ben Gamla, who without which,
"נשתכח תורה מישראל"
“The Torah would have been forgotten by Israel”.
This is because initially, the father would teach their child Torah, but all those without fathers would never receive a Torah education. As a result, the chachamim installed מלמדי תינוקות – teachers of young children in Yerushalayim ensuring that any youth could go and learn Torah. However, this did not prove to be a conclusive solution, since all those without fathers had no-one to take them to Yerushalayim. This resultantly led to the school system being expanded, whereby the local authorities instated teachers in every province. This too proved to have teething problems2, until Yehoshua ben Gamla ensured that teachers were placed in every district and town, and the townspeople were told to bring in their children, aged six or seven3 to be taught by these teachers.4
The Shulchan Aruch writes quite clearly that the onus is on the father. If a father is unable to teach his child, (which is the accepted norm nowadays, as most are either working or in full time learning,) he has an obligation to pay for his son to be taught. One must ensure though, that this process is not rushed, and this approach extends to a child’s entry into school. The Gemara in Kesubos 50a explains that Rav Shmuel bar Sheilas, a prominent teacher of young children, would not accept students until they were of age six and above. From this point onwards, he would “stuff them like an ox”, meaning he would infuse them with concentrated Torah study. The Gemara goes on to reason that if a child attends school before this age, the parent will always have to carefully monitor his child’s health as the learning will no doubt weaken him. The alternative understanding dictates that the young child will learn so much that his peers will seek to be like him, and they will become weakened as a result. It would stand to reason therefore, that the child’s entry into school must come at the right time. Failure to heed this call, will risk harming the child’s natural development.
So, let's take a deep dive into specifics.
As far as Torah learning, there is no doubt that in either scenario, a Rebbe has to give all of himself to the child/children. The Gemara in Sanhedrin 19b famously writes:
אמר רבי שמואל בר נחמני א״ר יונתן כל המלמד בן חבירו תורה מעלה עליו הכתוב כאילו ילדו שנאמר ואלה תולדות אהרן ומשה וכתיב ואלה שמות בני אהרן לומר לך אהרן ילד ומשה למד לפיכך נקראו על שמו
Rabbi Shmuel bar Nachmeini said in the name of Rabbi Yonason, any person who teaches Torah to the child of his fellow, the Torah considers it as if he were his own child. As it says, “And these are the generations of Aharon and Moshe” and it writes further, “And these are the names of the children of Aharon”, to teach you that Aharon bore them and Moshe taught them, therefore they are called by his name.
Essentially, the Gemara is telling us that although Aharon’s four sons were biologically his, since Moshe taught them Torah they were also called by Moshe’s name.
With this in mind, one can make the argument that it is infinitely easier for a Rebbi to go "all in" if he only has one child under his care vs a whole classroom. However, this too can be countered by noting that chazal set up standards to ensure that the classroom does not exceed the requisite number of kids.
There is a predetermined teacher-to-student ratio within any one classroom. Rava famously ruled in Bava Basra 21a:
סך מקרי דרדקי עשרין וחמשה ינוקי ואי איכא חמשין מותבינן תרי ואי איכא ארבעין מוקמינן ריש דוכנא ומסייעין ליה ממתא
The number [of students that] teachers [have in their classes] is twenty-five children. If there are fifty, we install two [teachers]. If there are forty, we appoint an assistant teacher and [the teacher] is given support from the town [in order to help finance the cost of hiring an assistant].
It is clear from this chazal that a delicate balance is to be maintained at all times. According to most Rishonim, twenty-five would represent the maximum number in which all the students would receive the requisite attention.5 However, above this number, greater assistance is required in order to ensure that all the students have their educational needs met. Having said this though, the Tashbetz in his Sheilos Uteshuvos, argues that this maximum number of twenty-five students was not based on the students’ educational wellbeing but was rather established for the teacher’s benefit. He notes that a single teacher can actually be assigned a number of students well above this bracket of twenty-five. However, this is only provided it was fixed ahead of time. If this stipulation was not stated, then the teacher is within in his rights to limit the class to twenty-five, or to forty students providing he can hire an assistant. The provision of class size was fixed in order to allow the teacher of a large class the right to ask for greater assistance if the number of students in his class is too much for him to manage on his own. Yet, if the number of students amounts to less than twenty-five, the teacher cannot make this claim.6
1 As related by Rabbi Shimon Finkelman, 5 Great Leaders, (Mesorah Publications Ltd – 2005), p. 329
2 The Gemara there relates that youths of sixteen or seventeen who lacked a Torah education were sent to these provincial teachers. However, those students whose teachers became angry with him would tend to rebel against the teacher and would then leave.
3 Six if they are of healthy body and mind, and seven if they are weak – See תוספות ד"ה כבן, מסכת בבא בתרא כא.
4 Refer to the Gemara and also see SA YD 245:5
5 Refer to Rambam in Hilchos Talmud Torah 2:5, the Rashba, the Nimukei Yosef writes that with 25 students, "שדי להם במלמד אחד", also note the Ritva, and SA YD 245:15 – It writes there, "כ"ה "תינוקות מספיק להם מלמד אחד. However, it must be noted that both the Rosh and Tosafos ד"ה סך מקרי דרדקי כ"ה ינוקי view the number 25 as the minimum bracket. That means to say, a town is not obligated to hire a teacher unless there is a minimum of 25 students.
6 The Tashbetz says there - “And if they assign him more than 25 students up until 40 students, he is able to protest unless they give him an assistant. For example; a sharp-minded student who will hear from the mouth of his Rebbi and will then go and relay it to the students. And the townspeople are obligated to pay the wage of the student who is acting as the assistant. And if there is more than 40 students, even one more, we follow the ruling of the Rambam z”l and like the Ba’al HaTosafos z”l that the townspeople are not allowed to force the teacher to teach the children or even the assistant, but are obligated to hire another teacher, since it is no different to hiring a labourer tasked with light work only to then give him heavy work”.