If you point me to the dead links I'm happy to fix them as I come across them. The current site we have online, to the best of my knowledge really doesn't have many dead links at all (I can't think of any from the top of my head) but if you have a list of links/buttons that don't work please do share! I've set up email reports for all server errors and 404 which I do spend time working through. I've also spent literally hours and hours making sure all the old site URL's redirect to the new site in a logical way (try visiting an old PHP forum URL for example).
Working on the site concurrently with the old one as you suggest brings up a lot more issues, and is a lot more time consuming and difficult to develop. It just wouldn't have worked in our case at all. Not only that, it would never get finished, and sometimes you just have to decide a point to release. Appealing to too many people for comment also can do a lot more damage than good. Me and Ash both had a clear focus of what we wanted the site to be, and presently it's strongly heading that way.
Clean large releases for large websites really require a team of people, just me making it is really time consuming, I wasn't expecting it to be a 100% smooth release, but that's ok, that's just how things are. We have still executed a lot better than some other websites have in my opinion.
You are right that it's a work in progress, every website is really! Development on big and good websites never really finishes. I am confident in saying however, that the current website is qualifiably better than the old site.