English Heritage's statement at http://www.english-heritage.org.uk/server/show/nav.17513 says, "In the centre of the mound remains of plants and insects are preserved both at the top of the old ground surface and within turves used to construct the core of Silbury 1. These remains survive because very little oxygen has reached these deposits meaning that the normal process of decay by fungi and bacteria have not taken place, although the exact mechanism of preservation is difficult to determine and forms part of the current research."
Those remains have probably survived because the structure was stable and very little oxygen (or anything else) was able to reach the deposits until recently; this appears to no longer be the case due to collapses from the 1776 shaft and the ingress of water from (apparently) that source. In other words, after some four and a half thousand years the micro-climate within Silbury may (probably?) have undergone a possible dramatic change. The public is entitled, surely, to be told what precautions English Heritage have in place to prevent further degradation of the biological remains at the centre of the structure.
I'm not sure it's wise to assume that the only organic material at the centre of the structure will be, "...plant remains (which) are going to be tiny tiny scraps with identifiable bits, like pollen or seeds..." when larger organic items have already been found. Again, if something larger is found I think the public has the right to know now what precautions/facilities English Heritage has in place to preserve it - being organic, preservation is something that will have to be done right away, not left lying around for six weeks, or in the case of Silbury itself, sixyears.
It's true that most of the material being removed from the Atkinson/BBC tunnel appears to be backfill. English Heritage however are examining everything that is being removed in case it does contain something of interest (even English Heritage manage to get some things right ;-)