Yup, that's the danger of reading too much into some threads...
Do well in your academic studies and be on target for the typical offer and you, like anyone else, can be a competitive applicant.
Does your school ever send any students to Oxbridge? It's not a problem if they haven't but it can be a bit of a help if they have.
In addition TSR really is one of the best places to get help with a Cambridge application. Not least because we have a number of actual Cambridge Admissions staff regularly on here helping in threads or hosting their own thread where you can ask about any aspect of applying to Cambridge.
Also, when you get to Year 12 you could apply to the Cambridge Shadowing Scheme where you spend a week in a college during term time and get to really experience what it's like to be a student.
https://www.applytocambridge.com/shadowing/apply/NB. Cambridge has more state schoolers than private school students.
Fundamentally Cambridge is looking for good people, not good schools.