I figured out what the problem was:
I copied all of the rows from the Microsoft Works Database spreadsheet, and then pasted them into a plain text file. The data appeared as tab-separated values. At the end of the text was an extra blank line. I deleted the empty line at the end, and again copied all of the data. I opened the spreadsheet in Google Sheets, deleted all of the old data, and pasted the new data...and the cells were filled in just fine using the rows of new data.
An extra blank line that was included in the copied data must have been confusing the Google Sheets program!
I'm not sure what they changed, as my previous method used to work just fine, but for now, if anyone is having this problem, they can just do this extra step as a workaround, and it should continue to work as usual.
Removing the blank line was not necessary. Simply pasting the rows into a text editor and then copying it again was enough to allow the rows to be pasted properly into the Google Sheets spreadsheet.
Also copying all of the rows from the Microsoft Works Database spreadsheet and then closing the program brings up a window that says: "Save large clipboard contents? You placed a large amount of information on the clipboard. Click Yes to make it available for other applications after you quit Works." Yes or No. Clicking Yes and then pasting the rows into the Google Sheets spreadsheet worked fine after that.
So basically copying the rows from the Microsoft Works Database spreadsheet is in some special or unusual format that doesn't seem to match the Google Sheets format, but a simple paste and copy in a text editor trick seems to fix the problem.
Who knows why?