The monarchy continues on because most people favour it at this point in time, as far as I can see. A lot of people feel quite sentimental about it: there are some advantages to the current system in terms of continuity, attraction to tourists, national feeling about state occasions and so on.
So, George is likely to be Head of State one day. His family can easily afford to pay for his fees. Lots of wealthy families (and some not so wealthy, on scholarships) pay for the top independent schools. I can't see why the royal family shouldn't choose this route, given that we still have independent schools. Since we are a democracy, presumably the survival of these schools is the choice of the electorate, who could have chosen, if they wished, to vote for more left-wing governments and the closure of these schools.
In this situation, Eton seems quite a good choice. George's dad seems to have been happy there, and it isn't too far from home. There are security advantages, as pp have discussed.
This might not be how we would organise the higher reaches of government, or the education system, today, if you started from scratch. It would seem obvious to have as much equality as possible in the education system, and a democratic republic. However, these institutions have evolved over time in the UK, it seems to be what most people want or are willing to tolerate, and I think that's kind of okay.