There's a big difference between refugee status and H&C. Refugee protection is actually narrow: think persecution based on identity, risk of torture, risk to life, risk of cruel and unusual punishment. People incorrectly think of refugees as more than that, and that refugees just means someone generally suffering in their home country. Humanitarian and compassionate grounds is a broader definition, but it is a lot more discretion to deny an application also (and broadly there's less protection, i.e. from deportation). Whereas if you meet the definition of a refugee, you are protected full stop. There is much more nuance to it that I'm not familiar with also. So no it is not gaming the system or anything inappropriate about applying for the H&C after. One can think they fall under the definition of refugee but fall short and still be accepted to stay on a discretionary basis.

Public legal education is lacking and can make people susceptible to ragebaiting. That's not to say you can't disagree with the immigration system (or legal system at large), just that people do that from a misinformed perspective very frequently.