Why not have an open source Covid track and trace app?

The NHS app (and as far as i know most other countries ones) is a complete mess, this isnt really a surprise as its been rather hastily thrown together and couldnt really be tested on the wide range of devices it needed to be.

so why not allow the fine minds of the net to create an alternative which actually works? if the app was open source a vast number of people could contribute fixes etc and all for essentially no cost to the government

the only reason i can think against it may be some sort of privacy issues but i dont really think that would be any worse than the current iteration.

so is it a good or a bad idea? opinions?