So the Gotha Programme said: "The German Workers' party demands as the intellectual and ethical basis of the state: 1. Universal and equal elementary education by the state. Universal compulsory school attendance. Free instruction."

Marx says:

"Equal elementary education"? What idea lies behind these words? Is it believed that in present-day society (and it is only with this one has to deal) education can be equal for all classes? Or is it demanded that the upper classes also shall be compulsorily reduced to the modicum of education — the elementary school — that alone is compatible with the economic conditions not only of the wage-workers but of the peasants as well?

So his first issue is with both universally-equal and universal-elementary education.

Marx opposes universally-equal education because the present day society, unequal capitalist society, is unequal. The early stages of socialism / lower communism have to recognize this reality, recognize that perhaps peasants and wage-workers have different needs ("why do i need to learn X when I will likely stay a farmer?") / abilities (hard to focus on abstract math when hungry) from the well-fed urban middle class, that access to teachers and instructions and supplies (etcetc) is very different from rural village to rural village. Equal education is either a falsehood or a reduction to universal elementary education.

Marx opposes universal elementary school because the elementary schools of the lower classes are shit, both today and in Marx's time. It's hard to comprehend from our modern viewpoint, but the "20-30 kids in a classroom one teacher who's had maybe 4 years of rushed training" model of education is 1. recent, 2. horrible, 3. still miles better than the elementary education of Marx's day. He would rather keep the current (i.e. unequal) education system and improve the elementary schooling as the material conditions of the workers, peasants, etc, improve.

Marx is also notably against free university and ambivalent on compulsory attendance and free elementary:

"Universal compulsory school attendance. Free instruction." The former exists even in Germany, the second in Switzerland and in the United States in the case of elementary schools. If in some states of the latter country higher education institutions are also "free", that only means in fact defraying the cost of education of the upper classes from the general tax receipts.

For Marx, free education is a way of the upper classes using general taxes to pay for the education of their managerial class, their scientists, their bureaucrats, their writers and artists, etc. Today we see for example university admission and graduation rates as well as who gets employed in their field varying by class, as nepotism is rampant in both, and inequality in terms of resources (housing, food, tutoring, etc) is still at play.

Instead of this universal education, Marx says:

This paragraph on the schools should at least have demanded technical schools (theoretical and practical) in combination with the elementary school.

I.e. "teach the children actually useful productive skills instead of abstract liberal education created by/for elites jerking off how good elite culture is".

Marx also equally opposes state and church influence on schools:

"Elementary education by the state" is altogether objectionable. Defining by a general law the expenditures on the elementary schools, the qualifications of the teaching staff, the branches of instruction, etc., and, as is done in the United States, supervising the fulfillment of these legal specifications by state inspectors, is a very different thing from appointing the state as the educator of the people! Government and church should rather be equally excluded from any influence on the school. Particularly, indeed, in the Prusso-German Empire (and one should not take refuge in the rotten subterfuge that one is speaking of a "state of the future"; we have seen how matters stand in this respect) the state has need, on the contrary, of a very stern education by the people.

He's fine with the state setting general guidelines, but otherwise he wants the government out of the schools, doesn't want e.g. teachers to be state officials.

Marx's stance on schools makes more sense when one knows that Marx opposes the abolition of child labour (my emphasis):

A general prohibition of child labor is incompatible with the existence of large-scale industry and hence an empty, pious wish. Its realization -- if it were possible -- would be reactionary, since, with a strict regulation of the working time according to the different age groups and other safety measures for the protection of children, an early combination of productive labor with education is one of the most potent means for the transformation of present-day society.

Rather than separating the parent-workers (still working 12 hour days and losing hands without pension) from their children, sending them into the hands of the state educators to be given a liberal education about how good liberalism is, Marx would see the conditions of working improved enough that it's not horrible that children are working with their parents, because in working with their parents they gain a real education in what they're most likely to be doing their entire life, a real education in what society is like. As above, Marx would like to see this supplemented with technical schools (theoretical and practical), but I get the sense Marx has little faith in liberal education (elementary or otherwise) as a result of his own experience and his daughters' experiences with it.