Can I add custom buttons to the Touch Bar without using external apps (e.g., Better Touch tool)?
You can use Automator Quick Action to open Finder or iTerm2 and add them into the Touch Bar. Unfortunately you can't add some widget without third app...
Exemple of quick action use :
Write the script code and add it to the action, something like
tell application "Finder" to open folder "Documents" of home(you can also choose a color and an image)
Save the Automator document. When prompted, enter a name for the service.
The Quick Action button should appear in the Touch Bar after adding the Quick Action button (see below)
From Apple support :
Use Quick Action workflows on Mac
If you have a workflow you use frequently—for example, adding a watermark to large sets of images—and you want to make it easy to get to, you can create a Quick Action workflow. It is then available from Finder windows, the Services menu, or the Touch Bar (on a Mac with a Touch Bar).
Create a Quick Action workflow
In the Automator app on your Mac, choose File > New.
Select Quick Action.
Specify any of the following:
Workflow receives current: The type of data used as input to the workflow. If you choose Automatic, Automator guesses the type of input you want to use in your workflow.
Note: Only Quick Actions that receive files as input are available in Finder windows.
Application: The app or apps you want to use to process your workflow. Each app must be open for the workflow to be available.
Input is: If the workflow receives input from URLs, addresses, phone numbers, dates, or email addresses, you can specify different input options.
Output replaces selected text: When the workflow receives input from Text or Rich Text, you can specify that the output replaces the selected text.
Image: An image for your Quick Action. You can select an image on your Mac or choose an image from the menu.
Color: A color for your Quick Action (visible in the Touch Bar only).
Add actions to your workflow.
To save your workflow, choose File > Save.
After you create a Quick Action, it automatically appears in the Action pop-up menu at the top of a Finder window, at the bottom of the Finder window preview pane in Column or Gallery view, and in the Services menu. You can also access Quick Actions in the Touch Bar by adding a Quick Actions button.
Add the Quick Actions button to the Touch Bar
To access Quick Actions from the Touch Bar, you must add the Quick Actions button. You only have to do this once.