You are correct. But a turn spent doing nothing IS a turn, whether you spend the turn intentionally doing nothing, or you spend the turn doing nothing because you were surprised.
The Assassin may very well still be hidden, and still gain advantage because of that, or by spending Inspiration, or whatever. A 3rd level ability is not supposed to be the "I win" button. It is an excellent ability, as written. Don't Panic.
The Assassin should go first, he likely has a higher DEX. But sometimes, the fates conspire against us, and even our best laid plans fail: yes, you surprised the big boss, but when you jumped out of the shadows he happened to be looking right at you (his Initiative Roll was higher than yours). It happens - and if you hadn't surprised him, he would have spent that turn smacking you around, getting away, or buffing. Your carefully executed plan didn't have quite the dramatic effect you wanted, but it still had an effect. Rejoice, and maybe plan on picking up that Alert Feat...