"Has anyone here ever built or maintained a Third System? What steps did you take to get there? Can you really "plan to throw one away" in practice?"
I've been brought in a few times to cleanup Second Systems gone wrong.
The steps I took are to be frank -- the system built by experts didn't/isn't working. Note that you're calling someone's baby ugly, and you are not liked for doing this.
And yes, we are throwing one way. We're throwing away the ineffective software that's in place. Note that you're saying that the previous software is a cost, not an investment. you are not liked for doing this.
I've been brought in other times to build a working system. Every design goes through all three phases.
You don't understand the problem domain (or the technology). The First System is a mixture. It has problem domain kluges and technical kluges. The First System always has both.
In some projects, management declares victory, puts it into production and users hate it because the problem domain kluges. Developers hate it because of the technical kluges.
"Phase II" starts. In a few cases, I have continued on past the initial klugy phase. Now, the real work begins. Fix the problem domain kluges -- dig out the lies the users told, find the business short-cuts and stupid non-solutions. Fix the technical kluges, similarly, requires rework, which requires unittests so that you can refactor.
Note that the problem domain kluges are the most destructive. Users say "it's just a simple 'copy' of the data" when it's really not a copy but a second reference. Users say "just give me a user-defined field" and then turn that into a magical key that does everything (badly.)
Throwing away the poorly-defined (or poorly-understood) problem domain stuff is possible. It takes a while for users to wrap their head around what they're really doing. They lobby for "state of the art" process definitions.
Eventually, the users realize they mis-defined their business or their relationships or their processes. That's when the Third System can emerge. In one case (so far), I've continued on to this level of problem domain understanding. We're discarding a bunch of problem domain "expert opinion" and arrive at a clear-eyed view of what really happens. It's simpler than the Second System and doesn't have the crap workaround in the First System.
Yes, we threw away two business models. This doesn't appear costly -- it's just analysis and specifications. But -- really -- the old PPT's and plans and what-not are now laughable because they're so out-of-date and myopic.
Yes, we threw away previous software versions. However, we're using a lot of open-source tools, so it isn't very painful to stop using one component and start using another.