So I have in hand a EZG that is going back to Woodward for repair (bad digital output), I hooked it up on the bench and took a look at the cooldown function. On this unit Cooldown logic is not in Logics Manager, but in trying a couple things it worked as you described, a cooldown is present in AUTO mode, and in MANUAL on a trip it opens the RUN output immediately. This particular EZG was on a CAT G3516 with an EMCP II, so functionally should be similar to your unit.
I no longer have a Toolkit developer license, just don't that many projects anymore where I need to customize a control interface, but it appears if you want to make the cooldown work in MANUAL you will need a custom .tool file that will allow access to the cooldown logic. As an option you could go into Logics Manager and write your own logic to develop a parallel cooldown function in MANUAL, but not sure that would really be worth the effort, and don't know how familiar you are with developing logic in the EZG.
I have to get this packed up to ship out tomorrow, so i don't have any more time to play with it. As you pointed out above, you "normal" mode of operation is in AUTO. In most of the sites I work with MANUAL is usually just used for testing and running after service work to make sure all is ok before going online.
And thanks to you and TugboatEng for the nice comments, is appreciated.
Hope that helps, MikeL.