You have a number of problems so you need to find solutions to match those problems:
I think the working governors find it very difficult to come into school during the day. So hold it in the evening or on a Saturday morning.
The school has had a poor track record of parents wanting to be governors. So ask people directly; people you know and are not necessarily high flying career folks. There are plenty of people who can do ideas and turning out and you don't have to be a fantastic career person to be a very good governor. I see your comments about the PTA but has anybody actually asked specific people? Perhaps some from there might be tempted at the end of their tenure to move to a governor role? They already volunteer and that's half the battle - getting people in who are motivated.
Also, we would be wary of parents who are over-invested in their child! You may well be excluding some of the most motivated people. With good, tight management Mrs X who thinks the provision for gifted and talented children like her Tarquin could be a great governor.
We are a village school So parish council, church, local scout groups, Lions, Rotary etc etc can all be approached. Put posters up in the library. Put them up in the GP surgery and any clinics. Get them up in the playgroup or nursery because if you start getting the message out early because the more times people see a message the more they think about it. Is there a parish newsletter? Can you get an article in that?
I believe in training people Great! What training is available? Do you have open meetings so people can see what is involved?
We are also in an area of the country where there are lots of dual income families and people commuting to London. They have so little time. Ok, so what can you do about this? When and where are meetings held? Are the agenda and minutes published and circulated in a timely way? Can people join the meeting remotely (by Skype, Facetime, Google hangout or another video conferencing method)?
It has been me, as a lone voice, calling for people with more time! It was getting pretty obvious people were struggling with commuting, self-employment and high flying jobs, not to mention family time! So change the way you do things as per above.
The others felt skills, ready made, and people like them was the best way forward and wanted to follow the new formula for school governing bodies. You need diversity, especially if the current "people like them" way is not working! The new formula is good and very useful but you do need a critical mass of people too just to get stuff done and share the workload. Most people are capable of being a governor with the right support and time. If the current way is not working it needs to change but doesn't need to be totally scrapped and thrown out the window. It is not an either or situation.