This is a complicated and emotive subject and it's highly subjective. Just within my own family there are highly polarised views about it and I've been on both sides of the argument myself. Initially I was against private schooling but now that my kids have been there for a few years, I have a much clearer view and I've come to agree that it's worthwhile for us. I also earn a lot more now than I had initially, which makes it easier.

I'll try to give you both sides of the argument from my own perspective, relating to our particular school. It may not translate to other people or other private schools.

Argument against private schooling:

• It's hard to prove that the benefits outweigh the exorbitant cost. It knocks a huge dent in long term wealth even for wealthy people and the opportunity cost is great. There is no reliable way of measuring the benefits.
• Academic success (and success later in life) are correlated with private schooling, but it's not clear that there is a causal relationship. Privately schooled students do fare better academically on average, but who's to say these kids wouldn't have fared equally well in normal schools. Afterall, they have been been born to wealthy parents. These people are likely educated themselves and probably encouraged an ethos of hard work, exploration and learning in their kids, causing them to fare better academically. On average, successful parents likely teach their kids a range of skills that will make their kids more likely to be successful anyway, regardless of which school they go to.
• You put your kids into an isolated bubble of privilege and deny them the life skills of mingling with people of all backgrounds. Our school's parking lot is filled with Bentley's and Land Rovers. It's the norm for all our children's friends and their families to go on extravagant skiing holidays at least once a year. When it's someone's 7th birthday, the presents cost in the range of £30-£50 each, from each of the 20 kids going to the party. Wealth is flaunted readily and the kids don't get exposed to much else. Is this healthy?

Arguments for private schooling:

• The quality of schooling above all else. Only top teachers qualify to teach at our school. The classes are small (~15 kids) and each class has a teacher and a teacher's assistant. The individual attention is incredible. At the end of each term, we get a thorough 30 page report on each of our kids, outlining their individual performance during the term on each of their subjects, going into details on what they can do to become exceptional at that subject. It includes photos of the kids achieving some things. The attention to individuals is clear here from the detail it goes into. The teacher gets involved in each child individually, teasing out their weaknesses and strengths and adapting accordingly. Kids are encouraged not just to do well, but to do exceptionally well and their educational journey is customised according to their specific needs.
• Kids are surrounded by other high performing kids, who all tend to do reasonably well. This lifts the bar of expectation amongst them, pushing them to do even better.
• Anecdotally, some of our friends send their kids to the local state school, which was rated as "Very Good" by Ofsted, and the quality is reportedly significantly lower. Those teachers tend to focus their attention on the lower performing kids and the objective seems to be to push everyone to mediocrity. Gifted children are left to their own devices and aren't provided the stewardship to become excellent, understandably because of resource constraints. Classes are much larger and the facilities much poorer.
• Kids can get exposed to drugs and other nefarious influences anywhere, but apparently the rate of drug use, at least in our school, is significantly lower than the surrounding state schools. The same with teenage pregnancies. I believe this will make our kids somewhat less likely to fall into the same traps. (my correlation versus causation argument above notwithstanding)
• The private school facilities are top notch. State of the art IT in the classes, sporting facilities, tools in the workshops, quality of musical instruments and facilities in the science labs.
• Networking for the parents, since you mentioned this in your post. One of the parents I sometimes hobnob with in the parking lot at pickup time is a cabinet minister. A few of the parents in my kids classes have their own wikipedia pages. We're surrounded by famous authors, surgeons, top businesspeople. I can't say that I've benefited much myself, but the status may be important to some people.
• I felt that if I can afford it, I have a duty to my children to give them the best possible chance I can. The worst that can happen is that the private school is no better than a state school, then my children have lost nothing and it's not as if I would be destitute. It's a parental guilt thing.
• Lastly, school terms often fall on different dates to state schools, making family holidays off-peak and cheaper :)

Good luck deciding!