I think two of the three sources you give don’t really pass my standard for reliable

You know what? That is fair criticism, and I acknowledge it. My sources are not as good as they should have been and that's my fault.

I did some actual digging this time, and I did find a real academic source. The source is a 2006 academic book called Crossing the Line: Vagrancy, Homelessness and Social Displacement in Russia that written Svetlana Stephenson, who's a Sociologist professor at the London Metropolitan University who specializes in studying Russian society.

Although it's a good read, I'm not going to ask you to read a 170 page book for an online argument. Instead, I'll give the relevant excerpt from page 95:

The true extent of homelessness in Soviet times remains unknown, but it has been claimed that there were about six million vagrants in the Soviet Union in 1989 (Starikov, 1991), while studies conducted in the late 1980s place the number of homeless people west of the Urals at two to two and a half million (Alexeeva, 1993). This estimate is based on the number of people detained by the militia for vagrancy and ‘parasitic way of life’. As the militia registered the person each time he or she was brought into the militia station, this statistic is very unreliable.1

Source: https://www.researchgate.net/publication/255728652_Crossing_the_Line_Vagrancy_Homelessness_and_Social_Displacement_in_Russia

If we use these figures as a rough estimate, we get an idea of the extent of homeless in the Soviet Union. The Soviet population in 1989, as per the1989 Soviet Census, was 286 million. If we use the 6 million figure for the whole country then that means that the homeless were 2.1% of the population. If we use the most conservative number here, which is the 2 million west of the Urals, that would obviously be a severe undercount as that excludes the Russian heartland, however that figure does seem to be more reliable so let's pretend that it's the figure for the whole country. If that's the case then the homeless would still be around 0.7% of the population.

Now, I did my own digging for the US figures as well. According to the Alliance Housing Council, the homeless population in 1988 was somewhere between 1.3 and 2 million. The US population as per the 1990 census was 248.7 million. That means that the homeless make up 0.5%-0.8% of the population at the time. Keep in mind this figures come from an extrapolation of official figures that came out in 1984, and they made the assumption that the homeless population would grow 20% every year. So this is likely an over count, but I still think it's more accurate than the official figures.

Source: https://www.nationalacademies.org/read/1092/chapter/2#3

So going by these figures, I think we can safely assume that the Soviet Union not only had a homelessness problem, but it had both a higher rate and a higher number than the US at the time.

These figures I am more likely to trust, because the research climate for social sciences in the US was a bit freeer than in the USSR.

A bit is really understating it. There was an absolute canyon in the levels of freedom found in the US vs the Soviet Union. At the time the US was arguably the freest country in the world while the USSR was the least free. The USSR had no freedom of the press, speech, expression, assembly, information, nothing. The one and only source of data was the government, and they refused to report the actual numbers because they feared them. The fact that we're in the dark about it now should be proof that such a gap was substantial.

That's dumbest argument imaginable. Drugs are considered illegal, drug addicts access to free rehab in most countries... does that mean nobody uses drugs? Because that's your argument. The Soviet Union had homelessness. Just because they made it illegal, that didn't make the problem magically go away, and just because they denied it that doesn't mean it's not real.