They may have considered this trade-off. Privacy-minded folk who are unaware of the technicalities will be similarly suspicious of an app whether it is made by the government or a private company. Sure, the centralised/decentralised difference is a major technical difference, but if people aren't aware, or don't understand, or don't care, but are concerned about privacy, they may not install the app regardless if it's made by the government or a private company.
Then the cost (a few less people who would have got the decentralised Apple/Google app, but not a government-made one) might be smaller than the benefit (richer data).
What's important now is that we raise awareness of this so we can pressure NHSX to change their plan. If enough people refuse to use a centralized system they'll change track.
Then the cost (a few less people who would have got the decentralised Apple/Google app, but not a government-made one) might be smaller than the benefit (richer data).