At least using a credit card you have some recourse to take if there is a problem. Not as secure as you'd think or like probably. The problem with this, is that now a third party company you know nothing about, potentially has access to your supposedly private data, because it's stored at least temporarily on their computers. This certainly is not ideal, and hopefully this will be resolved in the future, but for now it is a fact of life. Many of the larger companies likely run their own web sites, so as to keep 3 rd party companies out of the loop.
Many people today do order products over the internet without encountering any problems.