Why doesn't Google+ have a count like Facebook and Twitter do in the sharing options of a single post/page?

Google+ doesn't yet support a counts API that we can use (as of March 2014). As soon as they support it we'll add it. We chose not to downgrade to the default Google+ sharing button since it wouldn't match the style of the other counts at all. If you absolutely need the Google+ sharing number to be present we recommend you use a plugin instead of the native AddThis functionality. Keep in mind the sharing functionality works as expected, it just doesn't show the number of plus 1's for the page.