Air speed over the wings is the only thing that matters for lift. Ground speed is irrelevant. In the right conditions you can get a small plane down to Zero ground speed and still be flying just fine. I've done it and demonstrated it to students numerous times.
On takeoff, you're fighting rolling friction from the tires and any uphill grade in the runway. If runway length is tight, you carefully calculate aircraft performance. There could be situations where it would be better to take of downhill with a slight tailwind vs. uphill with a headwind. Terrain could be a factor too. Montrose, Colorado for example; we landed to the south and took off to the north.
The numbers are fuzzy now but it seems like we were good with up to around 20 knots of tailwind although it did affect aircraft performance and increased the takeoff roll distance. You just had to run the numbers and see.
Airliners and cargo planes in Part 121 service are certified under more stringent rules (part 25, transport category) than aircraft in general aviation (part 23) and must demonstrate safe performance in dire situations (such as an engine failure just as you lift off the runway at maximum weight).
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