By typing some "nonsense" at the top of the .htaccess file and not getting an error (ordinarily you would get a 500 Internal Server Error) it would seem that .htaccess overrides were not enabled on the server. So, .htaccess files were effectively disabled - which they are by default on Apache 2.4.
To enable .htaccess overrides (to allow .htaccess to override the server config) you need to set the AllowOverride directive in the appropriate <Directory> container in the server config (or <VirtualHost> container). The default on Apache 2.4 is AllowOverride None.
With the directives as posted you would need a minimum of:
AllowOverride FileInfo Indexes Options
FileInfo for mod_rewrite, Indexes for DirectoryIndex and Options for Options and related directives.
Although it is common (and easier) to just set:
AllowOverride All
Reference:
RewriteCond %{REQUEST_FILENAME} !-d
RewriteCond %{REQUEST_FILENAME}\.php -f
RewriteRule ^(.+)$ $1.php [L]
These directives are not strictly correct. Whilst they may work OK for the URLs you are testing, they would result in a rewrite-loop (500 error response) if you simply append a slash to your URLs (and there is no directory by that name), eg. /home/ (or /home/<anything>). This is because your condition that tests for the presence of the .php file is not necessarily the same as the URL-path you are rewriting to. See my answer to the following question on ServerFault for a thorough explanation of this issue: https://serverfault.com/questions/989333/using-apache-rewrite-rules-in-htaccess-to-remove-html-causing-a-500-error
Also, there's no need to check that the request does not map to a directory to then check if the request + .php extension maps to a file. If the request maps to a file then it can not also be a directory, so if the 2nd condition is true, the 1st condition must also be true and is therefore superfluous.
And there's no need to backslash-escape literal dots in the RewriteCond TestString - this is an "ordinary" string, not a regex.
So, these directives should be written like this instead:
RewriteCond %{DOCUMENT_ROOT}%{REQUEST_URI}.php -f
RewriteRule (.+) $1.php [L]
(RewriteBase should not be used here.)
You can further optimise this by excluding requests that already contain what looks like a file extension (assuming your URLs that need rewriting do not contain a dot near the end of the URL-path). For example:
RewriteCond %{DOCUMENT_ROOT}%{REQUEST_URI}.php -f
RewriteRule !\.\w{2,4}$ %{REQUEST_URI}.php [L]
(With this 2nd version, it does not matter if RewriteBase is set - it is not used.)
DirectoryIndex home.php index.php
You gave an example URL of /public_html/home (to which .php is appended). However, this DirectoryIndex directive allows home.php to also be served when simply requesting the directory /public_html/. It should be one or the other, not both.
