Nobody will freeze to death waiting as temporary shelters are always available for everyone, exactly for that reason.

We do still have around 3500 homeless, around a thousand of who are categorized as "long term" and not just someone that's couch-surfing at their parents or friends while waiting for an a new apartment, but as everyone in Finland is eligible for enough social assistance that they could always pay rent and buy food, the majority of those 1000 are people who either refuse out of principle or have such bad drug addictions or mental health issues that they spend all that money on those or trash the places they are given and get kicked out.
And for those there isn't much anyone can do, as unless you commit enough crimes to get jail time you can't be forced to go to rehab, get treatment, or stop living on the streets either.

Here's an info page about it in English from the "No Fixed Abode NGO".

How much of their homeless population dies waiting for that housing?

None. Used to be that happened, back in the 60's and whatnot. Puliukkos dying under bridges. But not anymore.

https://yle.fi/aihe/artikkeli/2020/05/16/50-vuotta-sitten-helsingissa-kuoli-kymmenia-ihmisia-kadulle-pakkaseen-vasta-se (you'll need to translate but it often works pretty well)

Most people never even go homeless. Even if you're a schizophrenic drunk, you'll usually have your own apartment or if it's really bad, perhaps live in assisted housing.

While we do have a homeless population, we don't have people living on the street. Period. Yes, it's also to do with the fact that overwintering would be somewhat challenging. The small homeless population we have still have shelter, they just don't have an address to call their own.

The ones I knew were mostly basically hippies living in communes and some hc alcoholists who basically don't want to leave the shelters, because their social life is there.

I've perhaps once or twice seen a hobo sleeping on the street. And I've lived here over 30 years and driven a taxi at night for several of those years.

Our shelters are never full, so if you actually got thrown out onto the street willy nilly, you could just ring up the closest shelter and go there. Or if you're suffering from addiction, check yourself into rehab, where they might help with both issues.

Don't take me wrong, these systems have several significant issues, and often fuck up in practice, but mostly when it comes to the point of actually being under ghe threat of homelessness, the bureaucracy somehow magically does start working pretty well. (The bureaucrats realise how much more issues it causes.)

I've got loads of complaints about Finnish bureaucracy in practice, horrible shit, but the systems are good on paper, and others should definitely try them.

I know for the exact reason you said they've had housing shelters that all homeless people can sleep at for free. I got the feeling they were very barebones. As for locations for homeless people idk about the stats but I would bet my ass the percentage of Finlands population that's homeless is miniscule because it's so hard to be homeless there. Also, the total population is so small, but Finland is pretty huge, I'm sure they have more than enough room to house a bunch of people.