I think the National Museum (and any museum) would consider being a place of deposit of last resort as being a duty rather than a prviligee. In any case, therir grounds for doing so comes from their charter:
"its objects (‘the Objects’
which are to be
the advancement of the education of the public:
(i) primarily, by the comprehensive representation of science, art, industry,
history and culture of, or relevant to, Wales, and
(ii) generally, by the collection, recording, preservation, elucidation and
presentation of objects and things and associated knowledge, whether
connected or not with Wales, which are calculated to further the
enhancement of understanding and the promotion of research"
On the question of whether there is any rights for public access to the results of archaeological work on private land, the answer is no, unless:
1 the results are of national importance and hence coming under the remit of Scheduled Ancient Monuments,
or
2 the landowner chooses to seek planning permission for a development which has an effect on the heritage (the nation's heritage) in which case the mitigation of the effects of the development involves making the results available to the public and the records constituting the preservation by record available in the long term through deposition in an accessible archive (such a requirement usually being written in the brief for a project).
Otherwise there is no requirement on making the records or results of archaeological work public, apart from the moral duty of every archaeologist worthy of the name.