Programme managers in general ?
Programme manager is a job function which is not specific to software. The term "programme" in it is not directly related to "programming" or a software artifact.
"Programme" is a term that means a long term endeavor, be it in politics (e.g. a new healthcare programme) or in industry (e.g. the Airbus A340 programme). In the software industry, a programme often corresponds to a family of products, and programme management is hence a part of product management.
A programme manager is then responsible to manage and coordinate all the many interdependent projects aiming to realize and contribute the long term objective and the larger picture (e.g. the different software components and their releases).
So yes, programme manager are still being used !
Programme managers according to your article
Nevertheless, there's an ambiguity in the article that you quote:
A programme manager (according to the definition I give above) will make sure that the overall vision and objectives are clear, and that the projects remain aligned with it. But in large programmes, he/she will in principle not be personnally involved in the writing of functional specifications. He/she might not even be involved in the approval of these specs. He/she "just" has to ensure that the many projects are able to get it right.
But your article describes a small team and a "programme manager" who'd write the specs. This seems not to be the "programme manager" as generally understood. It seems to be more a traditional project organisation, where the project manager cumulates the role of lead analyst.
This approach is also still in use, but I think that "project manager" is the name that most of us would give to such role.
However the trend, is to go more for agile approaches, with "product owners" replacing the traditional "project managers" in a less hierarchical way. The specs are then worked out collaboratively in form of user stories, the product owner playing the role of business advocate to validate the stories and the priorities.