Or you bring your capital ships to that system (that raises culture as well).
End of quote
Not quite.
Capital ships ONLY neutralize enemy culture, they do not produce any for your empire and do not add to its spread rate along phase lanes. You have to have the building for those.
Scenario: you own a planet (with no Temple of Communion) between your own planet (with one Temple of Communion) and an enemy planet (with two Broadcast Centers). They're spewing out twice as much culture, so your middle planet's allegiance is declining. However, you park a couple high-level capships in orbit around it. They push back 1.2 culture per second, showing away the Broadcast Center's evil propaganda, allowing your own media Temple's enlightened culture to reach and increase allegiance at that planet ( 2.0 enemy culture - 1.0 your culture - 1.2 your capships = 0.2 your net culture increase). If you had parked the capital ships at your other controlled planet, there would have been no effect.
Think of them as a shield that helps slow the influx of encroaching enemy planetary culture attacks.
-- Retro