Their solution (as documented here: https://support.signal.org/hc/en-us/articles/360007059752-Ba...) requires that you place the devices "nearby" and wait for them to detect one another. For whatever reason, neither phone ever "saw" the other one (despite them being on the same network and physically next to one another). I tried this about 10 times before I finally gave up. (iPhone SE / iPhone 12 Pro)
What made it more irritating was that you only get one "phone detection" attempt per account transfer process, and you're rate-limited on the server side, so I got soft-banned for 24 hours after several attempts.