Ok but let's not deify gov.uk. Yes it's very good compared to most government websites but it's not the peak of web design. They are so anally averse to any form of JavaScript or interactivity that often the web pages become quite tedious to use, e.g. when picking dates for payments, they don't have a "tomorrow" button even though that's what you want 99.999% of the time because that would need the dreaded JavaScript.
Personally I'd always want "today", never "tomorrow".
So now we have two extra buttons on the form.
And perhaps the extensive user testing that gov.uk do would discover that for most people this added UI complexity makes the form less usable than keeping it as simple as possible.
Today isn't an option for the form I'm thinking of (tax free childcare); you can't set up payments for the same day. Really you want "asap" but I was simplifying for discussion.
I agree the tax-free childcare form could certainly be improved (I've even written a script to do the three-monthly reconfirmation for me, see my latest gist, though that's policy rather than tech!), but you can set up payments for the same day, you put today's date in (that is what I do; it gives a "You've asked for a payment of £x to be made to [site] today" extra confirmation step). There's nothing in the service manual page that says they couldn't enhance this form with a button as you're suggesting, and hopefully one day they will, though you're always actually worrying it could get worse...