You're right @fideloper, I should've listed what my logs were showing. I was, however only checking the syslog but the laravel.log had some more valuable information that could've helped in resolving this issue days ago. I wasn't checking it because I thought that the application wasn't even being touched by the server's cron.
After a lot of digging I found that the root user apparently did not have permission to access the .sh file that I made as an alternative for crons to directly access the php artisan commands.
I had to install a mail server called Postfix to be able to dump the cron's more detailed error logs into this mail server in order to see what was REALLY going on behind the scenes on the server side.
sudo apt-get update
sudo apt install mailutils
I chose all the default settings instead of trying to set it up and had it dump the routed server notifications into a /var/mail/root.log directory to which it then showed me all the errors that were occuring.
From there I found that one of the many bugs was that the root user didn't have permission to read the .sh file which was fixed with the below example:
(*Note - Be sure to cd into the directory where the .sh file lives)
chmod +x cron-task-here.sh
Then, through a lot of trial and error, I found that you had to access the artisan with your crontab's HOME config set to:
HOME=/
and then have the .sh read as follows:
#!/bin/bash
php /path/to/artisan schedule:run
There were a few things I could've did better to more effectively resolve the issue but regardless, I appreciate all the help!