It can, if your bank supports Spectre (which apparently is a common API service for banks). It's a bit awkward, you have to sign up for a third party to do it, but worked well for the three months or so I used Firefly.

Here are the docs: https://docs.firefly-iii.org/en/latest/import/spectre.html

(For what it's worth, now I use Pocketsmith, a NZ-made Mint clone that I'm pretty happy with.)

I export transactions from my bank CSV and import it into firefly, which works fine for my usecase.

Speaking of which, does anyone know if EU's PSD2 will finally allow me to read the balance of my own bank account via an API, or do I have to subscribe to whatever fly-by-night ad-driven SaaS that'll get a deal with the banks and scrap their site?

I'm not trying to scrap bank sites, even though I could, because a) it's a PITA to get through the increasing amount of login hoops they deploy, and b) I don't want to risk even a small chance that they'll lock me out of access to my own money because I "hacked" their site.