Looks like now the English version works again: https://foreigntalentact.ndc.gov.tw/en/cp.aspx?n=D927ED39BDAE7478&s=DA2F7BC919B77E24
I called the “Ministry of Labor” contact under the gold card section (Mr. Huang (02)2380-1720). Someone else who received the call directed me to another person not listed on the website, who confirmed that yes she is the correct person to ask about this kind of progress questions (if the status is stuck at “Professional Review by Workforce Development Agency”). In most cases this status means they are waiting for the review result from another ministry. In my case actually I just could have waited another 1 or 2 days and would have gotten a message either way 
PS: Not sure if it’s cool to post unpublished names and telephone numbers here, so I’d suggest to simply calling the number above. Maybe nicely ask if the person answering speaks English and state your question, and then you’ll also be directed to the correct person. Everyone there was very kind and helpful - perfect.