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Atomic clocks measure time passing.

Seconds are a fraction of a day which is Earth rotating, and count 86400 seconds and then roll over to the next day, but Earth's rotating speed changes so how much "time passing" is in 86400 seconds varies a little. Clocks based on Earth rotating get out of sync with atomic clocks.

Leap seconds go into day-rotation clocks so their date matches the atomic clock measure of how much time has passed - they are time which has actually passed and ordinary time has not accounted for; so it's inconsistant for you to say "Unix time really is the number of seconds that have passed" and "does not include leap seconds" because those leap seconds are time that has passed.