@JasonRandall
Is there one quiz per section or is the same quiz used for multiple sections? I'm going to write like it's the same quiz and you're using differentiated assignments because that's the more common usage.
Setting assignment overrides can be done through the REST API, but it is a pain through the web interface. The batch edit of dates supports assignment overrides, but you have to have it already set up ahead of time.
The demand for what you're asking for hasn't risen to what Canvas needs to justify development. It's not glitzy or glamourous. It doesn't make the masses go "ooh and ahh". It saves a lot of time for some people, just not the masses. Even groups you would think have the numbers go unheard. Math and science teachers have been asking for improvements needed for our quizzes for years without movement. For other things, they develop what seem to be very niche requests that [seemingly] no one would want. I think it may have to do with pet projects or code that someone contributed.
If I had to do what you had to do, I would write API code. That's only because I know how to do that. If you don't know how to use the API, there's such a steep learning curve that it would be quicker to manually do it. If you do this every term, then it might be worth looking into or looking into alternatives -- like not combining the sections together. Then you can set a default time for each class and it works. There are trade-offs on that, though.
Before I knew how to use the API, I would use AutoHotKey (Windows app) to automate sending the keystrokes necessary to do what I needed. Depending on the situation, I might read the date from a spreadsheet or I might put the date in the clipboard and then define Alt+1 to send the first timestamp, Alt+2 to send the second timestamp, Alt+3 to send the third timestamp, and so on. Then it's a combination of Ctrl+V Alt+1, Ctrl+V Alt+2, etc. I might even consider adding the Ctrl+V to the hotkey macro. This is definitely where using keyboard navigation wins out over mouse clicks.
If I was forced to use the web interface, I would make heavy use of the clipboard to set the times. I would the quizzes in different browser tabs, then cycle through them. I would add a section to the Assign to and get the date set. Then I would paste the time from the clipboard. I would do this for each browser tab/quiz. Then I would go back to the first and repeat that information for the section section. For the first one I would type the right time and then copy/paste it for the others.