Stage 3 does not always happen. There are a few remarkable counterexamples, but most research software does not usually finish taking the form a product.
In my experience, what happens is that some prototype works quite well and gets used in the next project. But the next project is building a new prototype, not improving the previous one. After a few projects, the person who wrote that first prototype already left the department, but there are people who depend on it to get results. If enough time passes, you will have some piece of legacy code that very few people can understand.
The problem with academia is that, while the prototype is an excellent topic for an article, the real product, a maintained (and maintainable) software project, is not. And getting funding in academia to do something of no academic interest (ie. that cannot be turned into a paper) is very difficult. If the software is of no interest to some industrial partner willing to spend money, that code will just keep moving between the USB drives of PhD students, probably branching into slightly different versions, until someone is brave enough to start working in a new prototype and the cycle starts again.
In my experience, what happens is that some prototype works quite well and gets used in the next project. But the next project is building a new prototype, not improving the previous one. After a few projects, the person who wrote that first prototype already left the department, but there are people who depend on it to get results. If enough time passes, you will have some piece of legacy code that very few people can understand.
The problem with academia is that, while the prototype is an excellent topic for an article, the real product, a maintained (and maintainable) software project, is not. And getting funding in academia to do something of no academic interest (ie. that cannot be turned into a paper) is very difficult. If the software is of no interest to some industrial partner willing to spend money, that code will just keep moving between the USB drives of PhD students, probably branching into slightly different versions, until someone is brave enough to start working in a new prototype and the cycle starts again.