One major reason I use Dropbox (other than the fact that it has, for the most part, "just worked") is to get files from the cloud on an as-needed basis on iOS phones and tablets. Considering my home Internet connection is not good enough to handle outgoing requests (like a server), I still need an online location to store maybe 20-30GB of files at a low price (my Dropbox account is a free one with a lot of bonus storage) that can integrate with solutions like Syncthing or Librevault or anything else. What are the options for those?

Syncthing can be weird at first, LibreVault seems more like Dropbox but doesn't provide a server.

Try https://hubic.com/en/offers/ it was created by "The number 3 internet hosting company in the world" OVH works quite well if you don't mind the dated interfaces

If you're okay with spending a small amount of money, you could spin up a VPS somewhere and run Syncthing/Librevault on it. For $5-$10 a month, you should at least be able to get 20 or 30gb of space.

You can get a VPS with 30 GB storage for $20/year[1]. Payment options allow to pay $11 per 6 months.

[1] https://www.vpscheap.net/pricing.aspx Section “Budget Servers”

Can you offer "public shares" with that?

I see it has no mobile clients, that's almost a dealbreaker for my every-day use. Syncthing runs great on my Android.