Excel treats times and dates as numeric values; however, it is possible for cells to appear as either of those (dates or times) but actually be formatted and treated as text. In those cases, errors will occur if we are treating them as numbers. As a quick check, use an unmodified version of your worksheet (one where you have not performed the F2 operation) and go to some empty cells and enter =TYPE(B4) in one cell, =TYPE(C4) in the next cell, and =TYPE(D4) in the next cell. This will return the data types present in B4:D4. We know B4 has a staff member's name in it, so it is text, and text has the data type code of 2. Numbers have a data type of 1. You should have results of 2, 1, and 1. But it sounds as if you really have 2, 2, and 1?
Often when importing information from other systems, you will need to clean up the data and confirm that all entries have the desired formats. Typically, Excel will attempt to "interpret" imported data and treat it appropriately as text, a date, a time, or a basic number (the latter three are all numbers internally, although they have different displayed formats). Sometimes this interpretation does not work as desired, and often with exported data, some hidden characters can be attached to data that appears to be numeric, but the hidden characters can prevent Excel from interpreting the data as a number when it is imported. The process to clean the data and ensure correct conversion into the desired data types is not always straightforward.
Here is the first thing I would try. Enter the number 1 somewhere on your worksheet and copy it to the clipboard (again, use a bad version of the worksheet where the dates are being treated as text). Then select the entire column of apparent "dates" (that are being treated as text) and select Paste Special and choose the Multiply option in the Paste Special popup window. This forces a multiplication of all entries by 1, so Excel will automatically reinterpret the text-dates and numbers, multiply those dates by one, and then return the numeric value for the date...if you are lucky. If this works, then you will probably want to re-select these cells and reformat them with a short date format so that they appear as conventional dates again. Let me know if this works, or if the text-dates do not convert. And if this does work, then you will need to add these processing steps when the worksheet is updated.