Re Weymouth Quay: there was no closure proposal under the old legislation. By the time the relevant bits of the 1993 Act came into force (1/4/94), it had slipped through the cracks: as there had not been any regular, scheduled services for more than 5 years, the new closure provisions did not apply [see e.g. s39(1)(b)]. There are no provisions for retrospective closure but if there were, in the absence of any associated purpose (the ferry), I can't see there being any good - economic - argument for keeping/restoring it.
Coming back to the present, I doubt that there is any appetite for wholesale closures being pursued. Apart from the political considerations, they take up a lot of time and effort (but surprisingly little obviously additional expense, except for the cost of publishing notices in national and local newspapers) for, frankly, little benefit as would be the case for most of those mentioned above. The only real benefit of a closure is if it paves the way for something else, e.g. where a better station at a nearby but different location is being provided, or a big project like Thameslink 2000 (as was).