The point about HA CE's pay is that it comes mainly from tenants' rents. It is therefore relevant for the regulator, not the government directly, to oversee how they are paid and to take action if it is deemed excessive. Similarly for charities and the role of the Charity Commission.
HA CE's pay has risen markedly with the increased mimicking of private sector practices - heavily encouraged over the years by the Tories, who said behaving like the private sector was the way forward for public bodies (how their argument has changed).
Many HA Board members are now from the private sector where they are used to gravy train policies for top bosses. They naturally support higher salaries for CEs and restraint for the ordinary staff.
Despite what Shapps says, this problem will get worse under his watch because he plans to get rid of the regulator, the TSA, which has taken a special interest in top pay. Another example of how the policy of scrapping the TSA is just silly.