I think you are mistaking how private companies pay employees and how a publicly funded organisation pays employees. I don't care how much my bank pays people as it's the bank's money whereas everyone has to pays TV tax if you own a TV and people don't want to see it squandered. I can guarantee he wasn't forced to take the job!
I think you're possibly ignoring the simple fact that if you employ someone in a job with specific skill requirements, it tends not to matter if it's a "public" funded one, or a privately funded one.
You want someone with the experience and skills to do a job, you pay something at least within shouting distance of what they can get in the private sector, or you tend to get staff who cannot get a job elsewhere*....
The DG of the BBC was being paid less than most of the top guys at the other broadcasters.
IIRC the DG of the BBC has been paid less than the second or third guy at most of the other broadcasters, even before they halved the pay for Entwhistle.
There has been a certain amount of leeway in what people will accept for the prestige of having "worked for the BBC in X role", but that won't work very well when you are paying a third or less of what a newspaper editor can get, for a much more complex job than that of editing a daily paper.
Comparisons to the likes of the PM are rather moot, as to be the PM you don't need any particular skills, just to have been a popular politician (and one of the failings of our political system here, is that basically the PM is just the guy who leads the most popular party), especially when you consider the retirement package the PM gets, and how much they can make the moment they step away from the job (what's Blair charging now, isn't it something like 50k for an evening talk?).
Certainly many of the politician in the UK who have professional "qualifications" don't seem to know things you'd expect of someone with those qualifications (like all the supposedly qualified lawyers in the various parties that don't seem to know basic law).
*And it's interesting how many of the ex BBC staff seem to be able to walk into the same job at another broadcaster and get an instant pay rise.