I was thinking more like your A2 tab. This would require 2 sheets. One with just the unique dates, and the second like you have in A2. On the Dates sheet create a relation to link the date from the dates sheet to the date in the A2 sheet. The sheet with the dates would be the tab where you select the date. Once you click on the date, you will see the details for the date. This is where you can create an inline list using the relation and viewing as a checklist to show of the the names with a checkbox.
Another option is to just stick with the A2 sheet by itself and set a filter to only only show names where the date column is within Today.
There are a lot of ways to do this, so it’s pretty much your preference. My idea is going to create a lot of extra rows and exceed the free row limit pretty quick. Your original attendance tab is the most efficient, but you’ll have the problem with adding and removing names. If you wanted to get really crazy, you could create a column for name and a column for true/false and repeat several times in the sheet. You could then use the rich text component to show the name column value as a heading and then a checkbox component to record attendance in the true/false column. This won’t look as good though and may be a lot of work.