Btw, I just realized, I was using 5.11 before. Will updating to 5.13 be any good solution?
sudo apt update && sudo apt upgrade
should take care of that. Zorin OS is now using the 5.13 kernel. You may have just not run your updates, yet.
I wonder if your Elitebook is using Elantech...
It still shows 5.11.0-46 even after running the query in the terminal 
In terminal, pelase run
dpkg --list | grep linux-image
And relay that output here.
Please open Software & Updates and ensure that "Download from:" is set to "Main Server."
Upgraded to 5.13. Still not detecting trackpad and still not loading synclient 
I had doubts that it would help. For a 2015 model, if it was a driver in the kernel, it would have most likely been supported by 5.8 or 5.10.
Although we know you have already tried some of the tips here:
https://www.maketecheasier.com/fix-touchpad-not-working-linux/
You might read through in case there are some things you have not yet tried.
You may also try testing LiveUSB of other distros and see if any of them have support for your notebook touchpad. It's worth checking.
I found that this issue does happen in some of the hardware. And if it does, it carries itself to all distros.
In the attached thread some people have found the solution, but I cannot understand half of what is written 
We can ignore the bit about the kernel, since you have already been on the 5.13.
It suggests ading a Grub parameter:
sudo nano /etc/default/grub
Arrow key down to this line:
GRUB_CMDLINE_LINUX_DEFAULT="quiet splash" and change it to look exactly like this:
GRUB_CMDLINE_LINUX="i8042.notimeout i8042.nomux"
Once done, hit ctrl+x to exit, then the y key to say Yes to save, then the enter key to save the configuration and exit.
Now, update grub (essential step):
sudo update-grub
Reboot and test - run the command to see if your device is even detected.
cat /proc/bus/input/devices
Or xinput list
Thanks for all your support @Aravisian. Seems like I have to try some other distro as Zorin is not ready to detect my trackpad 
Except if @Aravisian you can guide on this specific solution mentioned in the thread. I hope this will be last time I will be bugging you 
Did you try Aravisian's this answer?
Yes, completely.
And it didn't work?
Aravisian summarised the solution of the thread that you referred to in your above posts.
Yes. But I just checked that this solution also did not work for the guy who posted this. Then he mentioned another solution involving hid_i2c. I am not sure how to execute it.
I can't help you with that but in case you decide to switch distros...Go for Pop! OS otherwise for a Debian based distro that is not Ubuntu based.
In all honesty, as I have paid for Zorin, I will prefer getting this solved (though the official support is pretty slow). Also, with the modern web, OS now has a very limited role compared to browsers. If I have to make a switch, I will definitely first try the Lite Pro and then will see where to move forward. But I won't be needing a switch is this issue get's solved.
You are already using 5.13.0-27 if you are fully updated on Zorin OS - I do not know if you are ready to compile and build a kernel from source.
You may try the module probers in any case:
modprobe i2c_hid
modprobe i2c_hid_acpi
The other suggestion is
sudo nano /etc/modprobe.d/blacklist.conf
Paste this line at the bottom of the file by using the arrow keys to get to the bottom and creating a new line with enter key
blacklist elan_i2c
Then tap ctrl+x to exit, the y key to say yes to save, then the enter key to save and exit.
Touch-pad is not my specialty, but there's a PPA with the latest kernel - if everything else fail you could try this.
