I am told that professors will always accept me even if I had average
or below average marks because I am bringing my own fund for this
degree yet this whole notion seems false.
I think you've hit the nail on the head right there. I have no personal knowledge of any PhD program in the world, outside of zero-reputation programs begging for students/diploma mills, where having mere funding will get you in. Now, maybe if you have the funding to have a new building built in your honor you could get the President/Chancellor to put in a good word for you, but I don't think that's the kind of funding we are talking about. Even then, though...
Maybe I'm reading this wrong, but when I read over your question and responses I feel a sense of feeling entitled to other people's time and resources. I have funding, admit me; I have a question, answer me; I want to be admitted, work with me; I want to meet with you, make time for me.
While we are all entitled to basic human dignity and respect, and I feel that's a very broad line, that's really it. Any good professor has many things they wish they had more time to do - work with existing students, do research, improve their program, improve their courses, reach out to the community, write a book, maybe spend more time with their kids and family or indulge in a hobby, consult, serve the University in another capacity, etc.
If a professor you approach gets the sense that you think they are obligated to work with you for any reason, that you are entitled to their time and energy, the vast majority will shut you down cold with little hesitation; for them it will be a reminder that there are indeed people who deserve their time and they wish they had more time for those people, but you aren't one of those people so go kick rocks.
So it might be time you seriously considered how you present yourself, and alter your strategy for how to approach a prospective degree program. While you feel that you can perform at the levels the department demands, they don't know that - so you need to consider how they might see you and feel, and how you can best present yourself as someone worth taking a risk on. There are a number of questions and answers on this site on how to approach prospective advisers, as well as general "open letters" from faculty around the world giving their advice on how to handle such interactions as well. I'd strongly recommend you make use of them, and I think you will have a lot more 'luck' if you take good advice to heart.