EDIT(2): Apparently it was not a bug, but corrupted files. The Check Files option in the Playonline menu found quite a few files it had to fix, and apparently the quest were one or several of those files.
The mistakes of panic and paranoia, posting here, after all. Pardon the trouble, but thanks for reading.
Edit (1):
*Re-logging does not help, as it will attempt to initiate the cutscene again.
*When I was stuck, I had to ask my sister to call a game master, as I could not access menus or do /commands myself, I could only use the chat for chatting purposes.
*I have finished "Full Speed Ahead!", and Promathia & Zilart Missions having reached "The Last Verse".
Platform: PC, Windows 10
ISP:Telenor
Type of Internet Connection: Fiber
Internet Connection Speed:50mbps
Date & Time: 01.06.2019 23:50 GMT+2
Frequency: Always
Character Name: Nuraki
Race:Mithra
World:Sylph
Main Job:Blue Mage
Support Job: Dancer
Area and Coordinates: Upper Jeuno, Chocobo Exhibit (G-7)
Party or Solo:Solo
NPC Name: N/A
Monster Name:N/A
Steps:
1. Finish ZM17 "Awakening"
2. Finish "Full Speed Ahead!"
3. Walk or Run into the Chocobo Exhibit in Upper Jeuno (G-7)
4. Prepare for the cutscene for "The Flying Machine of Eld" to initiate.
5. Because the cutscene does not initiate, you are now frozen until a Game Master teleports you out.