You roll a stealth check against the creature's passive perception. You remain hidden until you attack or make noise.
I'm out of patience to explain stealth rules, I recommend that you read this rule.
This is not how the stealth rules work. If the GM determines that conditions are proper for your PC to hide, you can make a hide check, which is contested by other creatures perceptions. Usually this is passive perception, because there's not an active action to perceive, but if a creature does take an action to search, your stealth check is compared to their perception roll, not their passive perception. Corner case, but if you're insisting others read the rules, it's good to be extra clear. Going back to proper conditions, you must be "not clearly seen" to be able to hide. This is not just an entry condition, it's a continuous one. You cannot, for instance, successfully hide and then remain hidden as you cross an open floor.
Then, you remain hidden until the conditions change. This definitely includes attacking, but not necessarily noise. If you make sufficient noise, the GM may rule you're no longer hidden, but there's levels there and it's a judgement call by the GM. Also, as noted above, if you lose the cover or concealment that is allowing you to hide, you're usually automatically discovered. Again, there are corner cases, but this is the normal case.
So, to the point of the post. The cases where you, a sorcerer, will be able to have the scene be such that you can approach unseen and undetected, achieve a vantage where you can observe the enemy without exposing yourself, and then employ your ability to cast without sound or movement will be uncommon, not standard. The stealth rules actually suggest this to be true. Further, you're arguing that you can get expertise in Stealth, which would require a feat. That would slow your progression in DCs and spell attack rolls, reducing your effectiveness as a sorcerer. And, you'd still have to MAD into DEX, alongside CHA and CON, just to maintain reasonable stealth checks. That's a lot of build resources just to win a "nuh-uh" contest, resources better spent on warcaster for the sweet Concentration buff or just straight increases to CHA. The claim that the rules work how you want them to rather than how they do doesn't really help much, either.
This ability just isn't a big deal. It's situational -- only those situations where concealing casting is very beneficial rather than just interesting. I'd be interested in the build in my games, to see it in action, but I have very clever players and I'm not the least bit concerned about this.
