Opening Crawl Not Playing

Assuming your not hitting the esc key at anytime once you create your character there is no setting that I found that would disable that.  I just created a new Sith Warrior last night and the opening text that crawls with the backstory played just fine.

I would try a couple of things.  First I would go ahead and add swtor.exe and launcher.exe to your anti-virus / firewall safe / exceptions list in case they are blocking or deleting files.

Next I would run a repair in case there are missing files:

Non-Streaming launcher - Log into the launcher but don't click Play. Check for small letters "ST" in the bottom left corner. If this is NOT present, With the new launcher after login , select the username pull down on the top right and select settings, then press repair.

Streaming launcher - If the letters "ST" is present, close the launcher, right mouse click on the game shortcut, select open file location and delete the BitRaider folder. Then start the launcher.