You running some weird addons stopping it? (or it is once again VPNs being blocked?)
It's probably the privacy settings or something related to that. I'm using
(security and privacy focused Firefox fork) in particular which has a lot of fingerprinting protection, and with it comes uBO, but I'm not using any other addons. The site does work over every VPN connection I've tried, thankfully.
I just uploaded and downloaded a file via transfer.it using Firefox (with uBlock Origin enabled)
Uploads work on my end but not downloads, interestingly. Once I click either "Download all" or "View content"->"Download" the page reloads to the beginning.
You know you could install a 2nd browser to download from there

(Brave / Opera / Vivaldi .. )
Yep, I have
(security and privacy focused Chromium fork) installed and use that just for transfer.it.
I could reduce some privacy settings in LibreWolf to see if I can get the page working, but since it works just fine in Helium I'm going to stick with that. IMHO more important to me would be if the jDownloader team could some day add support for the site, but I'm not complaining that I have to use Helium, just sharing my experience.