As with journal article and dataset publishing, it is a joint responsibility managed through a contract. An agent acting on the publisher’s behalf captures metadata and registers DOIs; and maintains the availability/functioning of those. The publisher is responsible for and motivated to maintain the availability of the article/dataset and accuracy of the metadata, and to provide updates about that to the agent. The publisher pays the agent by some mechanism - not always directly.
@dshorthouse and @hardistyar have expanded on this with some notes about the allocation of responsibilities for each part of a digital extended specimen in the sub-topic on structure and responsibilities of a #digextspecimen.
Who makes landing pages depends. Actually, many collections already making landing pages. Motivation and return on investment are important to consider. If collections-holding institutions want their data to be more widely used, to be more relevant to society, economy, etc. so they can receive more income - either directly through use of data or indirectly by public funding then new responsibilities must be taken on.
A robust agent/publisher contract would contribute to avoiding undesirable drift. But also, DS have their own lifecycle. Although this is associated with the lifecycle of the corresponding physical specimen it is also quite independent of it. We can expect presentations and uses of digital extended specimens to evolve in new directions.