I want to use the old clock applet in the MATE panel, instead of the clock in the indicator applet. Therefore, I don't want Ayatana indicator to display the time.
I disabled "Enable this Indicator on login" for the Date and Time entry in Ayatana Indicator settings, but it continues to show the time.
After disabling the "enable this indicator on login" button, have you tried logging out and logging back in?
Failing that, open up Startup Applications (Brisk menu -> Preferences -> Startup Applications, or Classic System menu -> Preferences -> Startup Applications). That should open a dialog with a list of programs set up to start running every time you log in. Look in the list for an entry like Ayatana Date & Time Indicator, and if you find it, uncheck the checkbox to the left of the entry. Then log out and log back in again.