Isn't the passage also saying, however, that with the "abolition of the present system of production", that public prostitution, the prostitution we are all familiar with, will also be abolished? I can see how someone would read this and conclude that "Prostitution should be abolished under socialism." or even "All forms of Sex Work should be abolished under socialism". Since, it seems, that what is being argued in the passages, is that these forms of prostitution only exist as a result of the social relations of capital. That because people are not provided a means of subsistence, and must labor for those means (food, shelter, etc.), some have or are forced to, exchange sex for those means, placing them in the most exploitative situation. In that way, the argument about the "Family" is the same argument about "Prostitution", especially considering that sexual assault within the marriage was exempted from sexual assault laws (and in some places still is).

It would seem that the Bolsheviks believed this position, which was, prostitution exists only because of the social and economic inequalities between men and women.

From Selling Sex in the City: A Global History of Prostitution, 1600s-2000s

Just as in France, the Russian state enacted regulations in 1843 that allowed for tolerated prostitution in cities as long as prostitutes were registered with the police and submitted to weekly medical examinations. This position, and the accompanying legal entity of the “public woman”, remained until 1917, when the Provisional government overthrew the system of regulated prostitution. This paved the way for the Bolsheviks after the October Revolution, who perceived prostitution as a prime example of the structural inequality of men and women, and its eradication as an important step on the path to gender equality. Thus, rather than reintroduce criminal measures against commercial sex or the medical supervision of women engaged in it, Bolshevik officials developed a system whereby they sought to rehabilitate and educate former prostitutes so they no longer needed to sell sex.

It's hard to say how effective they really were in my limited research. Following that paragraph, we have:

However, the system of labour colonies they enacted for these purposes proved useful vehicles for a punitive shift against women who sold sex in the Stalin period, during which time such women came to be deemed “social parasites” and “socially harmful elements”.

I've read some and skimmed other parts of this chapter. The chapter isn't specifically about the soviet era and prostitution, it is about the history of prostitution in Russia going back as far as the 1700s. Several times it is noted that there is limited data on prostitution during the soviet era, as the position officially was that it was eliminated as a result of the high employment of women.

Arguably, the influence of pauperization and proletarianization on the sexual economy was even greater in the early Soviet period, when the enormous upheavals of revolution, civil war, collectivization, and rapid industrialization produced an underclass of disenfranchised women who sold sex in order to survive. At first, during and immediately after the Civil War, many women (and men) left the cities, often starving for want of supplies cut off by war, and returned to the land where they could eke out a living. Thus for example, the population of St. Petersburg fell from 2.5 million in 1917 to only 700,000 in 1920. However, particularly with the beginning of collectivization and the first Five Year Plan in 1928, this situation was entirely reversed. The twin processes of collectivization of agriculture and “de-kulakization” (the confiscation of the land of “kulaks”) left millions of people without the ability to sustain themselves on the land, and they flooded the cities. This in turn led to a major housing crisis, as the housing in both Moscow and St. Petersburg was nowhere near sufficient for the number of people trying to live in the cities. Although we do not have statistics on the residences or social profiles of prostitutes working in Moscow and St. Petersburg in the 1930s until the ‘80s, we can extrapolate from the historical evidence of the extreme social dislocation, unemployment, and homelessness of the Stalin years that the need to sell sex for money may have increased in this period. Further, some historians have suggested that the social changes of the Stalin period changed the urban geography of prostitution in the cities, as “the housing shortage and the decline in private control over sheltered urban spaces appeared to drive illicit heterosexual sex into the streets, railway stations and carriages, restaurants, bathhouses and taxicabs.”

I should note that I have not investigated the sources here. It could be a reasonable assumption that prostitution had gone up as they suggest here. Especially if the argument is that social and economic inequality drives prostitution rates (which is a theme throughout the chapter). This next excerpt from the section entitled "Society and Prostitutes" I think gets to the core of what you're objection is to this framing, and I think it makes a strong case:

In the early Soviet period, this image of the prostitute as a victim was only strengthened, although now the primary victimizer was not a nefarious criminal or pimp, but capitalism itself which drove women into poverty and thus forced them to sell sex for a living. This interpretation of the causes of prostitution was indeed in accordance with much of the social data from the nineteenth century (as evidenced by the discussions above of the prevalence of former domestic servants and illiterate women among the ranks of prostitutes). However, it also led to the corollary claim that with the transition from capitalism to communism, prostitution would per force disappear. As a result of this ideological shift, by the late 1920s prostitutes began to be seen not so much as victims of fortune but as obnoxious evidence of the failure of Soviet society to remake economic relations and build real existing socialism. In the 1980s, under the influence of perestroika and glasnost’, Soviet commentators admitted what had gone unmentioned since the 1930s: prostitution continued to exist in the Soviet Union. Its continuation proved an analytic problem, however: in a society purported to have shed the vestiges of capitalism, the old explanation of prostitution as a result of economic inequality raised thorny questions. As historian Elizabeth Waters noted, most commentators solved this by reworking the classic Soviet understanding of prostitution as an economic problem and labelling it a moral failing instead; specifically, amoral failing on the part of the prostitutes themselves. This shift set the scene for the explicit condemnation of prostitution as an administrative offence during Gorbachev’s glasnost’ regeneration campaigns.

So this really I think signals the dangers of not thinking materially about the nature of Prostitution. On its face, the assertions that poverty and inequality drive up the number of women (and at times men, be they heterosexual or queer) performing sex work is I think statistically true. However, this does not mean that with a change in social structures naturally will eliminate Prostitution. What it would appear, based on what I've read here, that the Soviet leadership started off with an honest attempt at putting this piece of theory into practice, and over time, due to the many external and internal pressures they were dealing with as the first socialist state, they fell into dogmatism.

That is the risk of this assessment by Marx and Engels. Let us say we have our revolution and implement a dictatorship of the proletariat, we attempt to reshape the social relations within our nation, we have to ask ourselves the question: How do we make social changes that align with the theory that sex work is only an outgrowth of the social relations of capital? What if this part of the theory doesn't actually hold up, and even with new social relations, ones that are far less exploitative than the ones under capitalism, there still remains a population of sex workers.

The Soviets appear to have not accepted this idea, and turned on the women practicing sex work as a moral failing.

There is a whole chapter on Shanghai and Havana in this book as well, Chapter 22: Prostitution in Shanghai, Chapter 16: Prostitution in Havana, that I have not looked at yet, but I'm curious to do so.

I think the takeaway for myself here is this: Being ridged about the idea that, under Socialism, given enough time, all sex workers will have found new roles in new industries, is dogmatism at best. I think the realistic outlook is that in the near term, within my lifetime, even just beyond my lifetime, even if great social change will happen here in the Imperial Core, ushering in a socialist reconstruction of society, it will not be enough to dissuade people from Sex Work, and that it very likely is incorrect to even attempt to dissuade people. Presented with the opportunity, there might be a whole host of people willing to turn away from sex work for something different, but I can't say for sure that there wouldn't also be people interested in continuing Sex Work under new and very likely less oppressive and exploitive conditions.

disengage