I've always been surprised by the official homeless population count, but it turns out there's a lot more to it.
The department of HUD generates this ~771K figure from a "point-in-time" estimate, a single count from a single night performed in January. They literally have volunteers go out, count the number of homeless people they observe, and report their findings.
It's not hard to imagine why this is probably a significant undercount. There is likely a long tail of people that happened to be in a situation that night where they were not able to be counted (i.e. somewhere secluded, sleeping in a friend's private residence that night, etc).
Even if these numbers are correct, to my mind a "crisis" is still more characterized by the trend than the numbers in absolute. From the first link you provided, we saw a 39% increase in "people in families" experiencing homelessness, and 9% in individuals. A resource from the HUD itself suggests a 33% increase in homelessness from 2020-2024, 18% increase from 2023-2024. That is far apace of the population increase in general.
https://www.huduser.gov/portal/sites/default/files/pdf/2024-...
And even then, I would say many people would suggest that the change in visible homelessness they've experienced in the last 10 years would amount to "crisis" levels, at least relative to the past.
It's completely fair to argue that it is not in fact a crisis, but claiming that it is certainly not "baseless."
It's kind of wild that they pick maybe the coldest month of the year to do this. You'd think that would be when people are most likely to try to find some sort way of avoiding direct exposure to the open air even if it's extremely short term.