I understand the scepticism - we went through a lot of possible solutions before we found this one that actually works as a site search engine.
The site we are using it on does not have any password protected areas, so I have not tried it. We haven't felt a need to put it on the main site with the members only content so far. I have been very impressed with free-find and do believe it will work "to index parts of your site which are protected using the HTTP basic authorization scheme". Basically, it is able to index the password protected pages because you tell it the password info it needs to know to follow the links.
WHAT I DON'T KNOW is if WA uses HTTP Basic Authorization Scheme . . .
I have to say though, to get your money's worth with FreeFind, you really have to take the time to read their documentation and learn how to set up search parameters properly. It really is a powerful system - but it only does what you tell it to do . . . like all computer stuff, garbage in, garbage out. You have to understand what you want it to do and how to tell it to do it.
For example, for indexing all of the member sites in JAD, I had to learn how to set the starting parameters and the exclude parameters. The owners were very helpful in showing me the nuances I was missing at first - and that was all in the free trial.
Here is the documentation about indexing password areas - which they only recommend doing for low-security pages - http://www.freefind.com/library/howto/passwords/
It's definitely worth the free trial to find out if it fixes your problems . . .
(As far as WA, I don't know if they are open to WA integrating their software into the product or not, but you could always ask as a means of saving development time.)