Cannot move text using bracket along path.

The boxes are for threading text. If you have a text that flows from one line to the next (why you are getting new objects). The answer has been given, and it is to click on the line and not the square. (I understad you interpret it as a handle which it is not it is an inport and on the right an out port, similar to what is found in a text frame).

Unfortunately, the UI is confusing and not clear on how to move text along a path. Thankfully, the solution is quite easy: as others pointed out, use the direct selection tool (white arrow). But instead of clicking on the white boxes next to the text along the path, click and hold on the black lines along the white boxes. You can then drag as needed.

As there are two black lines lines, you'll often need to move the one furthest away from the text to open up the boundaries of where you can drag the text. Then click on the black line nearest the text and to move it.

When doing this for the first time, drag beyond the first black line (you moved) to understand what happens once you cross that line while dragging the second line.

I trust this helps.

Oink!

Taken me nearly an hour to do something that would have previously taken minutes thanks to Adobe messing with something that didnt need messing with. Very poor indeed.

I've had this problem for so long and can't seem to find a solution. It works randomly. But something I just noticed (or maybe just this project I'm working on for instance) is that it seemed to work when "Snap to Grid" was disabled. Maybe you could try that?

The type on path tool is a nightmare. I have always struggled with it in the past years. Once you finally get type on the path, trying to move it with the selection tool is utterly impossible. Half of the time the bracket(s) dont even show up. I am hovering for one hour trying to 1.) first get the bracket, then 2.) the arrow to show up. This is definetly a design flaw.

In any other software, you would simply point and click on the text, hold it and then just move it. Hovering to get something to appear is ridiculous. I first need to click on the selection tool (of course you first need to figure out that you need to switch from type tool to selection). Then you need to make the brackets appear (still no idea how thats done - sometimes they appear and most of the time they just disappear). When you finally have the right tool selected and you are lucky enough to see the brackets, then you need to figure out which bracket will actually move the text. Then you need to surpass the patience test of trying to get the arrow on hover to appear. Good luck, if for some reason both brackets are right next to each other or the path is in a cluttered area. 

Does Adobe even test these features with its user base?

If people need to post in the forums to get such a simple task accomplished, then there must be something wrong. I wish Adobe would spend more ressources on making tools and work flows in Illustrator more intuitive. I am not sure if there is such a thing as an "intuitive-design" - manager, but that is something the illustrator team urgently needs. None of the features should be able to pass the design stage without having been thoroughly testet by actual designers. I am sure it would also diminish support inquiries, so there is certainly something in it for Adobe too.  

As it stands, Illustrator feels more like a fiddely and awkward UI / UX from past ages than a modern app. Rather than holding on to cumbersome workflows and focusing on creating new features, I really hope that the Illustrator team will invest more time into making tools and workflows easier, more intuitive and more practical to use. 

You are trying to do this with the Text Tool. Use the Selection Tool (or Direct Selection or Group Selection Tool) instead.

Peter