I have home schooled, currently have two children in conventional state schooling, and also teach (in a school which most MNers would probably avoid like ther plague).
Firstly I would say that bullying is not necessarily more common than it was in the past. It is more openly discussed, but in many ways that is a positive thing because children are more able to say that they are being bullied, and schools are required to have strong procedures to combat bullying. That doesn't mean it doesn't happen - but my brother's bullying a generation ago was much less effectively dealt with by school than my son's bullying 5 years ago.
Secondly, I would say that bullying is possible outside school as well as in-school - at HE group meetings as well as in school, in Cubs or football as much as in the classroom or playground.
Thirdly, every child's education - whether at home or at school - has to develop in them the capability and confidence to combat bullying, as it is not a phenomenon unique to childhood. Whatever educational route you choose, it has to give your children the tools they need for their independent adult life. It will depend a lot on the child, the schools available to you, the family circumstances etc whether you decide that all those tools - social as well as educational - are best available to the child at home or at school.
Fourthly, I would say that you need to take every child, and every school, as an individual case. In my son's first (nice, small, village) school he was bullied and became so stressed by the chaos in his classroom that he became a selective mute. That was when I HEd him. In his current school (very mixed, big, in a large town) he has absolutely thrived. He has encountered the beginning of bullying behaviour - but the school dealt absolutely robustly with it and it stopped. He walks tall in school because he knows that he can deal with any problems with the help of staff. I could, and did, HE him to 'mend' him, but it is through successfully negotiating the school environment that he has acquired the tools that protect him against bullies.
If you WANT to HE, if the prospect excites you, if you are passionate about it, then by all means go for it. Just don't go into it because you are too scared to face any alternatives - explore all the options available, then make a positive choice.