The string is the point
Why Does a Kite Stay Up on a String?
A kite is a wing that never goes anywhere, because the string makes a headwind for it.
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Step 1 of 5
A wing only cares about air moving past it.
Which is why a kite in a gale and a plane in still air feel the same thing. A wing cannot tell whether it is flying through calm air or standing still in a wind. Either way air moves across it, and that is the only thing it needs.
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Step 2 of 5
A plane has to make its own wind.
An aircraft burns fuel to travel forward and keep air flowing over its wings. Stop moving and the flow stops with it, which is what stalling is.
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Step 3 of 5
A kite has the wind delivered, and the string keeps it there.
The string turns a blown object into a flying one. The wind is already moving. Without a string a kite would simply be blown along with it, and once it travels at the wind's own speed nothing flows past it at all. The string refuses to allow that.
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Step 4 of 5
Let go and it stops flying immediately.
A cut line does not send a kite higher. It drops, because the instant it can travel with the wind there is no air moving over it.
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Step 5 of 5
Which is why you can launch one on a still day.
Nobody has to wait for wind. Running backwards with the line drags the kite through the air, and most kites need only about ten kilometres an hour of it. Air moving past the kite is all that was ever required.
The short version
A kite flies because the string stops it travelling with the wind — a wing held still in moving air does exactly what a wing moving through still air does.
Try it yourself
Fly a kite, then walk quickly towards it and watch the line go slack. It sinks as you close the gap and lifts the instant the line goes tight again.