While you're not wrong that stealthy ≠ invisible, a proper mission design will render the aircraft effectively invisible to air defenses.

If you've got an active radar system you'll bounce signals off anything in the sky. Your ability to actually detect those things is based on the strength of the return and sensitivity/signal processing of the system. Big things can be detected hundreds of miles away, small things only tens of miles away. To protect some high value target you string together multiple radar systems to provide overlapping coverage. With enough systems you can have an unbroken wall of radar directing defending aircraft and SAMs.

Stealth lets a big thing (a jet) pretend to be a small thing in the view of an air defense system, essentially cutting the detection range of radar. This means your unbroken radar coverage that would work for an F-15 now has a bunch of holes because each radar can only detect an F-22 twenty miles out instead of two hundred. Your radar is also further compromised because the stand-off range of anti-radiation missiles is outside the range you can detect and intercept the jets carrying them.

Being able to see a stealth aircraft after it's fired a weapon to kill you isn't super helpful. A stealth aircraft can also fly through the artificial holes it made in your radar coverage and blow up the thing you're protecting and you only find out about it after the fact.

That isn't really how it works. Low observable aircraft flying in friendly civilian airspace generally have radar transponders turned on specifically to make themselves visible to air traffic control and prevent collisions. Those transponders are turned off for combat missions. And ATC mostly doesn't use primary radar any more so they don't even get skin paints on regular aircraft.