For a long time, the Chief left LA at about 8:00pm (8:30pm). There was usually a note in the timetable about dinner being served in the diner upon departure from LA.
If you know that a train is habitually late, I would suggest planning for that eventuality. Maybe you could take the Starlight as far as Santa Barbara, spend the night there, then take one of the Surfliner trains the next day into LAUPT and then onto the Chief. There is more to see and do in Santa Barbara (near the ocean beaches, nice town, etc.) than in the area around LAUPT. The main highlight is Olviera (?) Street, which is supposedly where LA began and has an old Mexican flavor to it. Very popular with tourists. Otherwise, there is not much to see anymore near the station and/or downtown. Although, you could take a train into LAUPT in the morning and take a bus tour around LA (Hollywood, etc.) and enjoy it that way.
Of course, that assumes money is not an issue. Still, when people plan trips and how much they will need for them, those are some of the conditions that have to be considered.
The same thing is true trying to connect from the Starlight to the Empire Builder. I would plan to spend a night in either Portland or Seattle and then catching the Builder the next day. Connecting from the Builder to the Starlight at Portland allows you a bit more leaway--the Builder arrives at about 10 am, and the Starlight leaves Portland at 2:25pm.
Incidentally, is there still talk about rerouting the Chief from its present Raton Pass Line to the BNSF mainline through Oklahoma and Amarillo?