If you're lucky you just set it to the wrong version, mine uses 10.3.0 (see below).

I tried running the docker container first as well but gave up since there are seperate versions for cuda and rocm which comes packaged with this as well and therefor gets unnecessary big.

I am running it on Fedora natively. I installed it with the setup script from the top of the docs:

curl -fsSL https://ollama.com/install.sh | sh

After that i created a service file (also stated in the linked docs) so that it starts at boot time (so i can just boot my pc and forget it without needing to login).

The crucial part for the GPU in question (RX 6700XT) was this line under the [service] section:

Environment="HSA_OVERRIDE_GFX_VERSION=10.3.0"

As you stated, this sets the environment variable for rocm. Also to be able to reach it from outside of localhost (for my server):

Environment="OLLAMA_HOST=0.0.0.0"

You can get different results, sometimes better sometimes worse, most of the time differently phrased (e.g. the gemma models by google like to do bulletlists and sometimes tell me where they got that information from). There are models specifically trained / finetuned for different tasks (mostly coding, but also writing stories, answering medical questions, telling me what is on a picture, speaking different languages, running on smaller / bigger hardware, etc.). Have a look at ollamas library of models which is outright tiny compared to e.g. huggingface.

Also, i don't trust OpenAI and others to be confidential with company data or explicit code snippets from work i feed them.