It is not possible to eradicate a species of plant while in its native habitat. It is not, ’pretty difficult,’… man himself simply can not do it!
With enough money, it is possible. Eradication of disease is much more difficult than plant eradication. But remember about small pox (well there are some in the lab)? It's at the last stretch of Polio. There are some species known from only a few populations even in the US. Nobody would do it, but what is you use herbicide there every year? If you are saying that it is impossible by collecting, then there is some truth about it. But this new statement isn't correct.
Of course I can deny the preposterous idea that population size reduction can/will have a significant effect on evolution of a species of plant. What kinds of nonsense is that? Show me the proof. (BTW: your reference to ‘Allee effect is to animals (goldfish), not to plants.)
Lance, you probably are not familiar with Allee effects enough, or you are misunderstanding. Any introductory ecology and conservation biology textbooks will have the explanation. There are several mechanisms of Allee effects. From ecological aspect, when the density is too low, pollination become near impossible (pollinator service become highly unreliable below threshold density). It is pretty relevant for insect (or other vector) pollinated plants. If the plant has genetic self-incompatibility, there may not be enough mating types left. In case of gametophytic self-incompatibility like tomato family, you need at least 3 alleles need to be maintained for any reduction to happen.
https://www.jstor.org/stable/20528879?seq=1#page_scan_tab_contents
From genetic perspective, in small populations, inbreeding depression can become severe. There is also mutational meltdown proposed by Lynch:
http://www.jstor.org/stable/2410432?seq=1#page_scan_tab_contents
This is not plant, but relationship between allee effect and eradication is discussed.
http://onlinelibrary.wiley.com/doi/10.1046/j.1461-0248.2003.00405.x/full
If you are willing to learn, you can find lots of theory, practice, examples of conservation biology with google scholar. Unfortunately, some of the papers are not publicly available (including the ones I linked above) and you need to get them via inter-library loan or something. Also many of these are not written in the way general public, which is unfortunate. I'm not an expert in conservation biology, so I'm not quite capable of explaining in a simple manner, but I can try.
Unless a study of this assumptive theory is based on personal and dedicated field work over a protracted range AND time-frame, with archieved type examples to show for un-impeachable verification, it can not be proven. (Your Ivory-billed Woodpecker a good example) It’s a bogus and usually ‘stupid-based’ process designed simply to produce an education degree or for a grade for a classroom project. The assumption of a ‘mutational melt-down process’ or your theoretic ‘stochastic fluctuation in environment’ (edu-speak) predictions, absent substantive field study is additionally such refuge for non-thinking persons.
Well, you are way behind in the current state of biology (I mean no offense here, since most people aren't trained in biology, and even people in the field has hard time to keeping up with the progress in the field). In this genomic era, there are lots of tools to look into what happened to the history of species from population genomic type data.
And it is a good point, conservation biology is complex topic since it involves something beyond science such as sociology, cultural studies, education, economics and policy. Even in the US where people aren't starving, educating people about the relevance of biodiversity isn't straight-forward.