I can't reproduce that.
Using *exclusively* the webUI, I first installed lighthttpd, then configure it, creating and specifying the server root through the Browse button and checking the other necessary checkboxes as shown:

Started it using the webUI and then accessing it though the webPage button (or on another tab on port 8080, as in
http://dns-320l:8080)

pages can be accessed OK and logs are there (System->Utilities, View Logs -- listed ad "access" and "errors" (yes, not a good name...) and no issues reported.
Also using https on port 8443, as in
https://dns-320l:8443/ works OK after accepting the connection after the usual https "take me out of here!" warning due to the self-signed certificate, which is expected.
What have you done differently? Editing configuration files?
Deleting the package might not delete configuration files that the user might have modified, that is probably the reason why it doesn't works after reinstall.
After deleting the package, remove everything under /etc/lighttpd, /usr/lib/lighttpd/ and /var/lib/lighttpd.
And at first stick to the webUI until you are sure its' working.