When you are using an online job site to find yourself a job, you want potential employers to find you and yet you do not want to be seen [by all and sundry, and especially by your current employer]! A definite conundrum?
Not quite! Suppose you get yourself a front man, say, your own personal career manager to act on your behalf? He could then discreetly (Though not like this!) make sure all potential employers are made aware of the fact that a person of your qualifications and caliber is available for employment. He could also check out these potential employers and give you advice.
Finally, after consultation with you, he could put you in touch with the potential employers of your choice so that you could take the discussion forward.
Now that would give you full privacy without reducing your options!
Why can't job sites do that for you? If they were really interested in making your life better, they should, don't you think?