Public school teacher, yes, private school teacher, no.
Quite simply, the teacher is a public appointed position of power. For many other public positions, it is also a criminal act when they have sex with someone who is their charge as a matter of their employment. Police officer, sheriffs, jail guards, prison employees, and all sorts of medical staff just as examples.
But that is also why I draw the line at private school teachers (even though prison employees and medical staff might be private employees). The public nature of the position of power is one of the critical reasons I think it should be illegal to sleep with your direct students.
There is also a secondary aspect of damage to the profession and to the public school system that comes into play; a level of damage that can rise to needing criminal deterrence.
That said, I think you still need to limit to students under current or recent direct supervision. That's why "damage to the profession" can only be a secondary aspect.
If a high school teacher sleeps with a student that teacher never taught and never supervised in an extracurricular, then it should not be criminal. A student at another school (e.g. elementary teacher sleep with a former student who is now 18)? Not criminal. So, if I were to craft this law (and such laws are often phrased this exact way), I would say a current or former student of the teacher who currently attends the school they teach at.
And it still should not be a sex crime.