Oh, yes, I think it's possible.
I don't think legislation is better. Much worse, really - and elimiating prostitution wouldn't be 'perfect' either, as women will still be sexually exploited. It'd be a tiny start, really, I think. 
Imagine it in terms of race. Imagine someone saying 'well, we could eliminate slavery, or we could just make it illegal to geld slaves - that'd help! And we could ban whipping ...' you could go on a long time, and it'd still make most of us sick to imagine that 'compromise'. And we'd all know that the elimination of slavery - the 'perfect' in your book - didn't in the least equate to 'good' or even 'liveable' for many, as there was still huge prejudice.
If you eliminated prostitution tomorrow, there'd still be a stigma against ex-prostitutes, and not against the men who exploited them.