I don't have sources on me but there are, iirc, a few factors at play:
It's really hard to keep track of homeless people. It can be done, but it requires constant interfacing with the homeless community. You can't just go out and do a headcount. You have to get to know them, know the community, and stay up to date. The funding just isn't there to do this and there are issues even when attempted.
The reasons people are homeless can summarize why it is difficult to keep track of the homeless. A lot of homeless people are either mentally ill and/or have completely opted out of society. It sounds weird, but there are say sex offenders who are homeless, or people who just become homeless as a way to opt out of obligations like debt or child support. It isn't quite intentional, but think of their mental state like you would someone who wants to commit suicide. Being mentally ill presents its own host of problems that are often individualized. These people are usually unstable to a degree they have pushed friends and family away. It's easy to think of their families as pieces of shit for not helping out, but I would recommend talking to nurses in a nursing homes. These people aren't always the easiest to track.
Organized shelters are often clusterfucks. I remember hearing about someone seriously injuring another person over access to an outlet to charge a phone. A surprisingly large amount of homeless people will have phones, these are usually the more temporarily homeless though. Bed bugs, fleas, roaches, are all frequent. As is terrible personal hygiene. Another anecdote I have heard is of constant shitting in the showers. Just imagine that. Then there are usually at least a few unstable people who make others feel threatened. Then you often get all sorts of agendas mixed in, like Christianity, or AA, or even just things like job programs can be very off putting. For the chronically homeless, all of these things suck. They aren't often just aren't capable of dealing with these things--which is a factor in why they're homeless. You can work at McDonalds and be homeless. Would you rather be homeless and work at McD's or just be homeless?
Homeless populations have something like homeostasis. Limited amount of beds, blankets, benches, and tolerance of homeless to go around. A lot of cities and counties will after a point just start rolling the homeless over to the next town. It reaches a breaking point where the homeless just naturally move on if they are able.