All right, finally was able to fix it. Here's a quick write-up:
Somehow I was unable to run pihole-FTL, but I was not sure why this was caused. Therefore, I checked my DHCP settings in the router (which were fine), and made sure to select the proper DNS Servers. All to no avail.
Then, I wanted to know what was actually wrong, so I ran pihole -d and since I could not upload the file and the token, started to dig myself. This then led me to an error among the lines of not being able to actually start pihole-FTL.
So I started to dig into pihole-FTL's debug which showed me that dnsmasq couldn't start because port 53 was already in use. So I ran
sudo netstat -nltup | grep 'Proto\|:53 \|:67 \|:80 \|:471'
to find that dnsmasq actually already was running on port 53. Thus I killed the process and started pihole-FTL, which, wonderfully enough, now started without any hiccups.
The only issue I now still have is that I cannot send a debug log:
[✓] ** FINISHED DEBUGGING! **
* The debug log can be uploaded to tricorder.pi-hole.net for sharing with developers only.
* For more information, see: https://pi-hole.net/2016/11/07/crack-our-medical-tricorder-win-a-raspberry-pi-3/
* If available, we'll use openssl to upload the log, otherwise it will fall back to netcat.
[i] Debug script running in automated mode
* Using curl for transmission.
[✗] There was an error uploading your debug log.
* Please try again or contact the Pi-hole team for assistance.
* A local copy of the debug log can be found at: /var/log/pihole_debug.log
Is there a way to solve this?