Radical plan to improve Whitehall

Whitehall on a Sunday morning
We took advantage of an encouraging weather forecast to take my toddler daughter into London for the first time on Sunday. We read her the Paddington Bear story on the train in, and after hugging the statue by Krispy Kreme, we headed down to Westminster to show her where Daddy sometimes works.
As luck would have it, we caught the end of some military/veterans parade or other, making its way up Whitehall. Judging by the black, yellow and red flags, I’m guessing it was Belgium Day or something. (Help! – JonW?)
Then something really odd happened. The parade continued up towards Trafalgar Square. In the few moments before Whitehall was reopened to traffic, it was one huge pedestrianised area. And it was beautiful. People went to look at the Cenotaph and ‘Women of WW2’ memorials up close, briefly lifting their status above that of a ‘keep left’ island. Quiet, gentle milling-about, as opposed to the usual game of Human Frogger. And no scrum by the Horse Guards’ sentry boxes.
Right now, Whitehall’s a right mess. Noisy, messy building work up both sides. The panels tell you it’s all about infrastructure improvements, and removing street clutter (?). But I can’t help wondering how much if it is because of the introduction of big security bollards outside some of the main departmental buildings. It would have been wonderful if they’d gone a step further, and closed the whole lot to traffic.
Would we get more done if we could adopt a bit of pavement café culture?