Is "Sequencing Breakthrough" applied only to a single player
You have been playing correctly. The effect applies to the next 'Discover A Cure' action, regardless of which player performs it.

The card is placed in front of the player for two reasons:


it remains in effect (the obvious and helpful indicator so that players can easily check whether it applies)
it is kept out of the discard pile (to prevent other abilities and card effects from being used to create a "stack" of Sequencing Breakthroughs)