Can a completely pinned piece still give check to the enemy general?
These answers restate the rules implemented by our open-source engine, which follows the WXF laws and is regression-tested against the official 110-diagram repetition casebook. Where an answer refers to a specific position, that position lives on the linked reference page, where the build verifies it against the engine before publishing.
Direct answer
Yes. Being pinned only removes a piece's own legal moves, it says nothing about whether the piece is currently threatening a square. If a completely frozen piece, one that has no legal move at all, happens to occupy a square that attacks the enemy general, it is giving check exactly as if it were free to move.
Two separate questions
"Can this piece move?" and "what does this piece attack right now?" are answered independently. A chariot pinned along the flying-general file, for example, still controls every point along the ranks and files it would normally reach from that square, its pin only removes the moves that would break the pin, not its underlying attacking power.
Where this catches defenders out
A defender can misjudge a position by assuming a pinned attacker is neutralised, when in fact it is doing exactly what it always did to every square it covers. Checking whether you are actually safe means asking what every enemy piece attacks, never whether that piece could currently choose to move.
Related questions
Does this apply to the flying-general pin specifically?
Yes, and to any other pin. A piece pinned against your own general by a chariot behind it still attacks everything on its normal lines.
Could a pinned piece deliver checkmate?
Yes, if its attack on the enemy general leaves no legal reply, the mate is real even though the piece itself cannot move.