Angeleno here:
Metro light rail (blue line, transfer green line to LAX, and shuttle to terminals) is the fastest (35-40 minutes) and least expensive ($2.50 per person) route to LAX on Saturday morning. Taxi is $46 and 1 hour or more. There are taxis lines at major hotels, but depending on the startpoint taxis might be a rare option (LA recently passed ordinance allowing flagging a taxi downtown, but most don't know or try) .
As for the speculated startpoints and blue line metro stations:
Disney Concert Hall or California Plaza are 3 blocks from Civic Center station
Marriott is 2 blocks from Pershing Square station
Racers sighted in Union Station are lost and likely boarded northbound tracks in error, which is easily rectified in the boarding area of Union station. However, the photo has them in the lobby area, which means they are likely headed to the taxis line, bus, or information booth.