Well, perhaps reading the question is an important pre-requisite.... I was talking about my and my peers' experience 20 years ago.
So 'lots of extra work' because there were entrance exams to prepare for, special classes for that and going down that route was a massive commitment. Teachers would not give the time to it if you were not very committed and a reasonable prospect. They actively put people off unless they were very serious about it. No-one would have been encouraged to do the extra work for no reason, because no teacher would have wanted such unproductive extra work.
State school, got people in every year, including in the same subject to the same college every year (oh, teacher just happened to know someone there or have been? Yes. This is partly why, in that subject particularly, they'd only coach and enter good prospects - quality control, so the school's entrants were always taken seriously).
If you are so competitive and with a traditional / conventional idea that Oxbridge is best per se, as well as being capable, then of course you would apply. That's about your family, not your school though, isn't it?
I thought you were asking why others take a different view and approach. Lots of reasons have been given. Many are based in values. You seem to place value on competitiveness for its own sake that many others do not.