Half the time I have to check with my wife what cares were done recently when my nonverbal kid gets fussy to try to identify why they're fussy. Logging makes it so instead of asking one can check the log, especially useful if the previous care person isn't available to be asked now

I'm not sure I agree. 10 or so kids lets them get a lot of practice socializing, with 3-4 caretakers. Without a tool like this, it's really easy to miss that nobody's needed to change little Mikey's diaper today - but that's information that can be important for them and the parents to know.

I also have kids in daycare, and while they're able to provide ample individual care, once you get past one adult to a specific set of kids and the kids swap between adults it becomes a much greater risk of missing someone's need because they can't communicate it clearly.
It also can make it faster to know when something last happened if you weren't the one to do it. If a kids fussy and the person who's been looking after them all morning has gone to lunch you can just look over and see that they got up from a nap recently, got a diaper change and that it's almost time for food.

It's not about cramming so many kids in that you can barely keep track and more about recognizing that you're caring for someone else's kids and so taking every reasonable step to ensure there aren't mistakes, as well as demonstrating to the parents that you've done so.
Our daycare has a list on the wall with the name of every kid in the room next to their evacuation plan and emergency kit (big baby/little baby rooms are connected. Sometimes they rebalance for lunch or just different activities which is when they update the list) I have absolute confidence that in the event of an emergency they wouldn't need to use the list, and also that they would still go down the list and look directly at each kid and also do a sweep while doing whatever response they needed.

As someone who's done a bit of work on procedures around systems and making sure they avoid negative outcomes I appreciate there being a process and checklist that's routinely followed.

Also, the digital lists are really more for the parents to be informed about what's going on. I know that I appreciate knowing where precisely they are in their routine when I do pickup.

It's an app. Do you actually think they're manually entering the time? The app is probably just rounding to the nearest 10 for display purposes. There's also a legal obligation to fill out an incident report.
You're caring for someone else's child and the law says if you felt the need to do something (ice pack) then the parents deserve documentation with timeline and response. Do you have a different criteria that's good for when a non-medical caregiver should need to tell a parent something happened to their kid?