"is it possible to create a shared set of tools to provide consistent, beautiful, and easy-to-use government websites?" Yup: http://gov.uk They even copied the colour scheme: https://playbook.cio.gov/designstandards/visual-style/#color...
Gov.UK is a terrible website. https://medium.com/@prenticemathew/ux-specialists-are-killin...
Now that we have gov.uk, whenever I google something about doing my tax, or registering to vote at a new address, or any other arduous governmental admin shit, I usually end up on a beautifully simple and focused gov.uk page, and I'm in and out in 2 minutes. That's what a government website is about.
I agree the homepage looks like a parked domain but whatever. Take a random page like this: https://www.gov.uk/alter-a-will-after-a-death
Look at how brief and clear the writing is. Look at how straightforward its URL is. Have a look at some more pages, look how consistent the design language is. This is exactly what a government website should be like.