I have installed pi-hole now 4 times from scratch on a complete newly downloaded and burned Rasbian Buster Lite image on a Raspberry Pi3. There is no other software running and the whole system is as clean as you possibly can imagine. I have shared every possible debug thing that could be thought of.
@Hasse It is regrettable that you did not succeed in successful deployment of pi-hole despite significant efforts on your part. I know how frustrating this can be, but allow me to share my experience. In my quest for privacy and security I have been using pi-hole for about a year now in a home environment on two RPi 3B+ devices running the light-weight Debian image of DietPi (headless), and had no problems whatsoever. I also use unbound on both systems, but nothing else. I opted for the DietPi-image because I am by no means a linux expert, but I am not afraid of using the command line.
A while ago I have switched both RPi's to the beta release/5.0 of pi-hole and had no issues with updating it essentially on a daily basis as the developers fine-tune this release which I closely follow, great work! Two weeks ago I upgraded both RPis to the Debian buster image of DietPi, installed from scratch and booting from a USB-stick. I subsequently did a clean install of the stable branch of pi-hole and then switched to beta release/5.0 on both systems. I also upgraded unbound to version 1.90.2. I did not encounter any significant problems in the process and the systems have been running smooth ever since, including periodic updates of the beta release. I have made donations in support of this worthwhile effort.
This is regrettable to hear. Of course, Pi-hole does work, there are many many people all over the world that will attest to that So this whole thing is rather odd.
Now you may not be the only person that has fallen over at this hurdle, some others may have.. but they may not have said anything. Though that is impossible to tell.
I hope that at some point in the future you can try again. Maybe it's worth trying to install it in a virtual machine or a VPS, to see how that works out.
From the extensive findings, I'd think that this is neither Pi-hole nor client related.
I think that something in your network (either your router or a dedicated firewall device) is blocking access to block public facing DNS port 53, so your router would be the only device allowed to connect to e.g. 8.8.8.8.
If you cannot configure your device (i.e. router or firewall) to allow DNS traffic to public servers, setting your router as Pi-hole's only custom upstream server could provide a solution, as long as Pi-hole is not set as your router's upstream DNS server at the same time (normally, a WAN/Intenet setting).