The solutions was in fact simple: we told our customers we wouldn't add any features or changes other than bugfixes. When you are telling the software is outdated and the support is gone they all understand it's time to move on [1].
Most of the time the customer also wants to get rid of the outdated website/webapp. So it's also a good reason to start something new.
[1] Ofcourse this doesn't work for all customers.
Or did you have a new version ready and at (near) feature parity with the old version? As the article describes, that investment can become very large when you put it off for too long. And the reasons they had to put it off were, at least initially, quite valid.
Anecdotal, there's a company that built their core product on Symfony 0.9x, and never upgraded. That was started around 6 years ago. The company is now worth 1+ billion on nasdaq, and still relies on Symfony 0.9x for the core functionality.
From a development point of view it sucks. Ask any developers.
From the business point of view, staying away from the kind of 6 month no feature s-bugs-only period you're talking about, is probably the reason it could beat its main competitor, and succeed the way it did.