Note that there is a partial implementation of this in D8554.

IMHO the X must be available outside menu. This is used a lot.

If this is anything like the modifier header changes it's going to be hard to find a good balance of exposing common operations, leaving room for the name, and keeping the interface intuitive.

One consideration: It's nice that currently the number of users button is close to the fake user button. That clarifies the relation between the. So personally I'm not too sure about putting the number of users to the left of the name.

IMO either the duplicate or the plus button should be exposed. Also, I'm not sure we support this in most situations, but maybe we can show and hide icons as the template's width changes.

A fact of the matter is that some buttons are used much more frequently than others, such as New Material (which should be called Duplicate Material) and Unlink, and therefore IMO should be always visible in order to be accessible with a single click (instead of hidden in a menu which would require an additional click for every Material you want to access)... And one button, the "Fake User", is basically required to be exposed at all times due to blender's unintuitive behavior of data-block users (if even with this exposed new blender users are confused, imagine if its hidden).
.
Scalability: I don't see why can't we have a dynamic solution. This problem only arises if the window size or Resolution Scaling (in Edit>Preferences>Interface) is smaller than a certain point. Solved by the proposals below of a dynamic change to the interface.
1- If the properties region horizontal pixel length is reduced past a certain threshold (be it due to window size or Resolution Scaling), the most commonly used FakeUser/NewMaterial/Unlink buttons are then replaced by a "Down Arrow" button, which opens the menu suggested in the task:

Example: "Fake user" button tooltip is currently:
"Fake User. Save this data-block even if it has no users".
But could be improved to:
"Fake User. Save this data-block even if it has no users.
A "user" is an object or mesh that is currently linked to the data-block (using it). If nothing is linked to the data-block, it is considered "orphan" and automatically deleted from the file on reload. If you want to preserve the orphan data-block in the file, you can give it a "Fake User". This prevents it from being deleted on reload.".

Example 2: the Linked buttons only appear if the user purposefully links a data-block, which indicates he knows what is a Linking operation, or if he opens a file with such a linked data-block.
The "Linked data-block" button tooltip is:
"Direct linked library data-block, click to make local, Shift+Click to create a library override".
Could be improved to:
"Direct linked library data-block, click to make local, Shift+Click to create a library override.
A linked data-block is a piece of blender data brought over from another file and that can only be edited there (changes to linked data are lost on reload). Making it local creates an independent copy of the data-block that can be edited locally. A library override creates a local editable version of this data-block that retains the linked property of non-edited parameters"

A suggestion is that after 2.5 seconds of hovering over a button, a floating "?" shortcut appears, that when hovered over opens a floating pop up with a brief explanation of what the button's operation does, with images or gifs if necessary, such as the examples above.
Maybe the 1 second hover tooltip could still be the one that is already there, and then after another 1.5s the secondary, more descriptive, tooltip appears, or the "?" icon for the tooltip.
Another option is that the "?" icon when hovered/clicked opens the browser with the Manual page related to the button operation.
The first Icon of the icon list if I'm not mistaken is already the Question Icon:

Also, I'm sure that "New Material" button should be renamed to something else. Because when you have an Empty material slot, you can press the gigantic +NEW button which indeed does create a new DEFAULT material with default parameter values. Yet when you press "New Material" icon you are actually creating a new material which is a DUPLICATE of the selected material, it's not a new DEFAULT material. Therefore these two operations should have different names and different tooltips. Ex: "New Default Material" and "New Duplicate Material" OR "New Material" and "Duplicate Material" OR "New Material" and "New Copied Material"

Unusual behavior: We can actually Alt+Click almost any button or field in Blender in order to perform the same change/operation for all selected objects, so having a modifier+click is not new. Having a specific Shift or Ctrl click for such a specific operation (linked libraries) doesn't appear to be so problematic, since a user who makes use of this feature is most probably not a new user. But for ease of use it could be separated into two buttons, "Make Local" and "Create Override", and once you click "Create Override" it's button disappears and only the "Make Local" remains.

The new UI design almost all commonly used functions are hidden (folding), which is really inconvenient. Please do not make change UI, this is not everyone likes, the 2.8 new toolbar and layout top bar, I almost never clicked. The industry standard is not the best, it was a few decades ago design.

Hello guys, I`ve noticed this change today.
I understand motivation.
But now I do not have even local overview, what nodes protected or not.

Orphan data will not show protected nodes until they have users.
And Blender File outliner do not show this info too. So I can`t manage datablock-protection fast anumore.

There is a thread for user feedback on devtalk: https://devtalk.blender.org/t/redesigned-data-block-selector-feedback-thread
Please use that, I follow it actively.

having to use drop-down menu, instead of one-clicking frequently used operations, really slows down the work

@Julian Eisel (Severin) okay, will do,