If the government is the client then by definition the public is the client, since the government is only acting on behalf of the public.
No, the government is the client, by definition, in a project that was supposed to be for the benefit for the public. That does not mean that the public is the client, not even indirectly. You don't become a client by proxy just because you are a taxpayer, that is not how projects work.
And yes, I shudder at the thought of uncounted number of backseat drivers trying give feedback. Letting more people look into something isn't trivial and it costs. What government projects should we allow full and instantaneous insight (aka absolute transparency) into and to what cost? What is an acceptable overhead cost for adding absolute transparency to every project? How much do you want to pay in extra tax just to humor this 'need' of yours? I rather prefer to pay less taxes and let the politicians heads roll if they get out of line, and that the money I pay in taxes go to healhcare, education etc. If we add overhead costs, we should make sure that we save money in the long run. Anything else is much more irresponsible than a failed project.
I do think we need transparency, to a certain degree. Absolute transparency only exists in utopia or where there is no regard for cost. Even if we have transparency things can slip through. Every project is a gamble.
If the UK taxpayers see that there has been something wrong going on, make 'em pay. Use your democratic vote.