Showing all records on a webpage
I want to know if I should delete this script and make a new one with prepared statements.


Yes, definitely.
This set of functions is just unacceptable, being so bad on so many levels. 


There will be dozens to hundreds connections made in your script, not just "2 connections" for starter, as your code will connect anew every time it executes a query. 
There is, as you rightfully noted, no support for prepared statements, which is a disaster alone. That escaping routine is based on the false notion that mysqli_real_escape_string's purpose is to protect you from SQL injection. Well, it's but a grave delusion.
error reporting is completely flawed. What a site user is supposed to do with a blank page telling them that a column not found?
a DBRead() function is silly. All its purpose is to save you a typing of two words. Seriously? And this trifle gain at the expense of making a gibberish out of the powerful language of SQL.


To show 2 records from a table (as it's stated in the code as opposed to the question title) you need only two lines with vanilla PDO, so you better stick with it:

$sql = 'SELECT * FROM publicacao ORDER BY id DESC LIMIT 2';
$dadosUN = $pdo-&gt;query($sql)-&gt;fetchAll();
?&gt;
&lt;?php foreach ($dadosUN as $UN): ?&gt;

//some html here to display the records

&lt;?php endforeach ?&gt;