They sustain it back up because you are dealing with Fleet footwork, support enchanters, and teams who know your kit well. Cait's entire gimmik is trying to kill in 1-2 seconds full HP. Land a net, with a trap behind them, and then Head Shot, and follow up after they land on trap with another Head shot. If you're trying to poke down, it doesn't matter WHAT adc you're vs; they will outheal simply because Warlord's Bloodlust *cough Fleet Footwork is back in the game.
You'll notice that the top Cait players aren't just getting Coup de Gras. They're getting Absolute focus, since Cait spends most of the game at high HP. They also get Fleet Footwork, where the "meta" players are getting Press the attack, even though Cait rarely gets in 3-4 attacks in a volley (even when trapped, Cait usually only gets in 3 attacks and has to make the most of it with a Headshot Net combo). They also Overheal vs Triumph, because Fleet Footwork is a shield battery.
Learn to not soley believe that Press the attack is good on all champs.