Take a look at the FAA's map of ORD.

If (as is now the case according to weather.com) winds were coming from the S-SE, then ORD was probably using 14 L & 14 R (with most heavies on 14R, since it's 13000' long).

Now, I don't know where AA's hangars are, but I'm guessing the South Cargo Ramp area, since (a) UAL's hangars are at the threshhold of 14R, and (b) either South Ramp or NE Ramp would be closer to T3 (but the FBO is at NE ramp). So, to get from the South Cargo Ramp to T3, your a/c would need to get a break in the traffic on the runway most suitable for heavy departures at just the time of day when many of the heavies are leaving for Europe. Then you have to "swim upstream" across all the traffic outbound from T5 to 14R (plus any traffic AA is sending to 14R), and get to your gate.

Also, as you can see here, there are some restrictions on the use of taxiway B by jumbos passing each other -- and B is the taxiway that loops around the ramp entrances.

Or, they're just doing this to piss you off!