Then I tried the below (which sort of works):

(progn
  (setq minibuffer-prompt-properties '(read-only t cursor-intangible t face
minibuffer-prompt))

  (defun turn-on-cursor-intangible-mode ()
    "Turns on cursor-intangible-mode."
    (interactive)
    (cursor-intangible-mode 1))
  (define-globalized-minor-mode global-cursor-intangible-mode
cursor-intangible-mode turn-on-cursor-intangible-mode)

  (global-cursor-intangible-mode 1))

Start emacs -Q and eval the above progn form.
The caveat is (and I cannot understand why) is that the cursor
intangibility does not kick in when you do C-x C-f for the first time! It
kicks in only from the second C-x C-f onwards.

The following steps try to explain what I mean (on latest build of
emacs-25):
1. emacs -Q
2. Eval the above progn form
3. C-x C-f C-x h (The read-only portion also gets selected!)
4. C-g
5. C-x C-f C-x h (NOW the read-only portion does not get selected as
expected)

So why is the cursor-intangible-mode not enabled in the minibuffer the
first time (Step 3)?

Also, would it be wise to enable cursor-intangible-mode globally? If not,
how can we have that mode always enabled in the minibuffer?


-------------------------

On the other hand, I have tested the below as an alternate solution
(probably better too as I am not enabling the mode globally) to always work:

(progn
  (setq minibuffer-prompt-properties '(read-only t cursor-intangible t face
minibuffer-prompt))
  (add-hook 'minibuffer-setup-hook #'cursor-intangible-mode))

If we want emacs to "do the right thing", should the above be made default?