How do pilots give the plane a flight plan to use when on autopilot?
Nowadays, it’s pretty much all done through computers and databases. However sometimes the pilots will input the flight plan manually into the plane’s computer. Regardless of the input method, it always gets checked manually by the pilots.
God forbid waypoint ELGER, which is supposed to be 100nm away from them, ends up being 1500nm away near Beijing.
Where are waypoints recorded on this flight plan sheet?
In the ROUTE section.

Do you do this through the FMC system? Hjvh
Yes it’s done through the FMC. That’s one of its many purpose.
Most airlines no longer use paper and assuming most airlines have mandatory efb on its aircrafts. The flight plan is not a standalone sheet that pilots receive but it’s OFP that’s a document that contains a lot of information including the flight plan which is typically on the 4 or 5th page along with the cost index value settings, speed, alt etc given with the waypoints it’s automatically send to the EFB by the flight dispatcher. What you have shared in the image is what ATC gets not the pilot.
The pilots have to typically enter the flight plan (like speed, alt restrictions etc) they have received into the FMC (data link with fms enables loading waypoints) and if they wish to make any changes they would have to be reported to the dispatcher and atc. In most cases when a pilot wish to make changes or decline the flight plan they are given it’s usually negotiated btw the dispatcher and pilot.
For more insights you can take a look at this thread
