webspeechapi | Web Speech API Examples for Gecko | Speech library

All I want is for the browser to output the text of the tag that I am hovering over.

You are starting from a legitimately good intention, but in fact except if you are making a special game or an innovative interface, it's a bad idea. Accessibility on the web simply doesn't work like this. You'd better try to conform to standards like WCAG to make your site accessible.

Several reasons for this, at least two big ones:

• Finding elements to be spoken with the mouse is a possibility, but isn't the common way to navigate on the web.
  • Blind people generally don't use a mouse because they don't care about the placement of elements on the screen, can quickly get lost or miss important information that way. They just need to have them appear logically when navigating with tab, heading by heading or by another mean provided by the screen reader.
  • For partially sighted users, using the mouse to read elements below the cursor is of help or not depending on their vision, but for the same reasons as blind users, it's often just a complementary help; and screen reader software have the feature built-in.
• Screen reader users have their preferences about language, voice, rate, pitch, etc. fortunately they don't need to set them for each site they visit

So, unless you are making something really special or new, prefer stick to largely used means to access your site.

But it does that after speaking out all the contents of the document first. I think it does that because it catches the document first before I can reach the element.

This is probably because of event bubbling.