Ah, I think...
 

Hmm, the statement

Application.WindowState = xlMinimized

did not result in all windows getting minimized. The
following does minimize all Excel workbook windows. But I
still do not know if it speeds things up.

Dim wkb As Workbook
Dim OldDispAlerts As Boolean
Dim w As Window

OldDispAlerts = Application.DisplayAlerts
Application.DisplayAlerts = False
For Each w In Application.Windows
w.WindowState = xlMinimized
Next w
For Each wkb In Workbooks
If Not (wkb Is ThisWorkbook) Then
wkb.Close SaveChanges:=False
End If
Next wkb
Application.DisplayAlerts = OldDispAlerts