@hax sorry I missed your first post; it works without a driver in the default configuration because the PiTFT is supported natively and the driver is included in the default distribution. By setting dtoverlay=pitft35-resistive you’re loading the driver for that screen. As yours is different that driver isn’t working and we need to get you running a different one; one of the ones they’ve provided on GitHub.
Just so you’re aware, the pitft35-resistive relates to the .dtbo file that lives in /mnt/boot/overlays. If you look in the GitHub repo at what those install scripts are doing, they’re copying a .dtbo file and setting some configuration variables. You can try copying the file that relates to your screen (I am guessing it’s tft35a-overlay.dtb into the /mnt/boot/overlays folder and renaming it tft35a-overlay.dtbo as is happening on line 5 in this file; assuming that’s the correct one for your screen. You would do this on the host OS.
Hope this helps!