Just a small information, Facebook requires everyone to be atleast 13 years old in order to create an Account (in some jurisdictions, this age limit can be higher.) Creating an account with false information is a violations in their terms. This includes accounts registered on the behalf of someone under 13.
so, How to Remove 'Message' Button on your Facebook Account?
Step 1:
Click on 'Edit Public Details' on your Facebook Account. Basically it was located at your Profile page when you've logged in. By the way, I am using Facebook Mobile on Mobile Browser.
Step 2:
Click on 'Edit Your About Info'
Step 3:
Navigate to 'Basic Info' then Click on 'Edit'
Step 4:
Then Click on 'Birth Year'
Step 5:
Then Change your age to be 14 Years Old. So basically change your age and identify yourself as 14 Years Old in order to remove your 'Message' Button on your account so strangers who you do not know will not be able to message you.
And also, make it 'Only Me' so no one will see it. I mean, it's on you though if you wanted to, but I suggest make it private. And Save it.
Step 6:
If you have others accounts, make a check on the account you've wanted to removed its 'Message Button' and see if it worked. But in my case it worked. (This January, 2021)