Their reasons for "why it doesn't make sense" are pretty lame and have no weight.
Actually, they are fairly well aimed. I suspect you didn't entirely understand what their reasons were actually addressing. To note, the original email made two primary claims. The first is that the very act of preventing perspiration causes cancer. All your points are in relation to that claim. The second claim is that there is a chemical specifically in antiperspirants that causes cancer.
All the reasons that the site presents as to why the original claims in the email are false is related to that first claim; that the act of preventing perspiration is causing cancer. Admittedly, if those reasons were not addressing that claim, they would be "pretty lame and have no weight." But as they were, they must be considered in that light.
1) The "reason" that the site provided was in reaction to the email's claim that antiperspirant prevents toxins from being sweated out. Even if we assume that cancer-causing toxins can be sweated out of the body, that would not lead to an increase in breast cancer because any toxins that would be sweated out would still get sweated out, just in different areas. Your claim that not all toxins are sweated out is a red herring; the issue is if ANY toxins can be sweated out, not if ALL toxins can be sweated out.
2) The "reason" that the site provided was in reaction to the email's claim that antiperspirants cause cancer because the unrelated toxins that antiperspirant prevents from being sweated out get deposited in the breast region. If there is a general increase of toxicity in a body, then every area that interacts with that toxin would be at an increased risk. The email essentially states that these toxins pool in lymph nodes, which means that one should have an increased chance of Lymphoma throughout the body, not just around the breasts. The email's claim requires a poor conception of how the lymphoid system works.
3) The "reason" that the site provided was in reaction to the email's claim that antiperspirants cause toxins to be deposited in lymph nodes near the breasts which then cause breast cancer specifically. The toxins would be flowing away from the breasts, not towards them, so the toxins would be more likely to affect some other area further down the line, not upstream of where they are introduced. It is like claiming that dumping nuclear waste in a river mid-stream will poison the source of that river but not the termination of that river. In order for the toxins to affect the breasts, the lymphoid system would need to be operating in reverse.
4) The "reason" that the site provided was in reaction to the email's claim that breast cancer occurs in a particular section of the breasts that have a high density of lymph nodes, which relates to the above three points. The reaction is that no, breast cancer does not occur only in those areas that the email had claimed. This is significant because if the cancer causing toxins were related to the lymphoid system, areas without access to that system should not suffer similarly.
So the webpage basically claimed that antiperspirant doesn't do what the email claims it does, that any increase of toxins should effect the whole and not just the part, that the lymphoid system (the supposed career of these toxins) does not flow in a manner that would transport the toxins to the supposed site of mutation, and that sites unrelated to these deposits of toxins suffer similar effects.
To be fair, the site only debunked the email's claims regarding the mechanical effects of antiperspirants. If you will note, it didn't address the issue of if the chemicals in antiperspirant cause cancer themselves. But that is then addressed in the statement that the NIH (the biomedical field's deity, essentially) is unaware of any conclusive evidence linking those chemicals to breast cancer.
A simple PubMed search should carry the day, in this case:
Abstract from the Bulletin du Cancer (french medical journal) stating there is no relation Abstract from the Eastern Mediterranean Health journal stating there is no relation.The best you have is abstracts like
this one which state that chemicals in antiperspirants have the capability of causing cancer in breast tissue, but it specifically notes that research actually needs to be done on if application to the skin could cause breast cancer. That is why the website claimed the NIH is aware of no conclusive evidence linking the two. The best you get is that the chemicals could cause cancer but there has no indication that the chemicals can travel from where they are applied to the breasts. Indeed, the research indicates that the chemicals cannot do so.
The website, and by proxy Radical Dreamer, is rejecting something because there is no evidence to support the claim and indications that the claim are false. Citations are nice, but citations only make research easier. Research should still be done by the individual.
But here is the link to PubMed and a link to PubMedCentral (the former is much older and has abstracts from around the world, the other is newer, has full articles, but doesn't have too much yet). Go knock yourself out.
http://www.ncbi.nlm.nih.gov/pubmed/http://www.pubmedcentral.nih.gov/