Private investors are pension funds, even just individual people. I mean you and anyother individual can go to companies like M&G and invest in hedge funds or anyother investment you want. The fund managers themselves are just individuals, and yes they benefit the financial 'in' people, but that could be you if you do a bit of research into who is managing what fund and what their past performance was like. All that information is largely in the public domain.
The BoE buying bonds from the market forces demand up, reducing the cost down for the goverment to sell more bonds. When they come to sell them it'll be at what ever the going rate is at that time, the effect to decrease demand and pushing the interest rate up for the government. Not the other way round that you're trying to argue. QE has the effect to reduce the cost of borrowing for the government, whilst also increasing money supply in the system at the detriment of increased pressure on inflation.