Selfhosted

Besides the time and responsibility of running the service, your biggest cost is going to be storage and maybe even transfer. None of the cheap VPS have big storage or bandwith, so I would not offer a service like this if I were you.

How much storage you want? Do you want any specific feature beyond file sharing?

How much experience do you have self hosting stuff? What is the purpose of this project? (E.g. maybe you want a learning experience, not using commercial services, just need file sharing?)

VPS is not that much, especially if you aren't storing a ton of media. Digital Ocean and Hetzner are good places to look. This will also prevent some networking headaches you're likely to have hosting it "on prem".

If part of your reason for doing this is to involve the kids in the process, then it's better to do it locally. Someone in the org has or can find an unused desktop computer that you guys can have fun with for pretty much zero cost. You will probably have trouble trying to connect to it from outside your network though.

i'd say start small. do the the webpage on some old hardware, maybe a wiki. content consumption things that would be uncomplicated for the group to adopt. avoid things that would mean managing accounts for other people early on. a wiki or some static page using something like modocs will be easy to run off a decent internet connection at the building. low bandwith usage and low traffic.

if your goal is to degoogle group, nextcloud could be helpful for the organisers. maybe if you have success on the simple sites you can get people on board with some hardware for a small nextcloud server. but dont plan on opening the next cloud up to the kids. thats a world of risk you don't need to open up.

As someone who has no real experience with Nextcloud: Do I 'need' it, when I already have a NAS with Synology Drive running on it, being accessible through Tailscale?

Would you believe Oracle OCI? They have an always free tier, as in you never pay. You need a valid regular credit card. At first I thought it was for a slow x86 instance, but it includes Arm hours equivalent to 4 cores, 24 GB RAM, 10TB of transfer a month, I think 200GB storage. Divide it up for an nginx reverse proxy in front of it, or HA Proxy if you are feeling ambitious.

Just a word of warning

Nextcloud is very finicky and can be prone to breakage. That's not the say don't use it but be realistic about the amount of work needed to maintain it.

Honestly I would just go for gsuite or office365 simply because they are less likely to break on you. It sucks that Nextcloud is a huge monolith but it is what we have.