I do not really know what a default text box is. I'm not sure why that command is in the right-click menu. I suspect that it is a carry-over from the menu versions of Word.
As far as I know, the only time the default text box is used is when you use vba to insert a text box. In the menu versions you would tell Word to insert a text box and it would insert the default which you could then modify. Now you are give the text box
gallery. Setting a text box as default does not change the gallery.
Text boxes are Building Blocks these days. I can assign a name and category so that a particular one shows up at the top of the list.
(Both names and categories are displayed in alphabetical order.) Setting a text box as default does not change what shows up in the list. You would save in the Text Box gallery to have it show up as a text box when you use Insert > Text Box.
Adding your text box to the building blocks gallery also allows you to add default text. Saving as default does not.
You can also create a keyboard shortcut for one and can use the F3 key together with the name to insert one, just as with other building blocks.

Automated Boilerplate Using Microsoft Word
"Text box" as a term is used in other ways in Word. The above is about the introduction of a shape that contains text.
The other uses include a text formfield or content control or a text box control in a UserForm. If that is what is being discussed, it is important to specify. Most experienced Word users mean the shape when they talk about text
boxes.