A6 - V6 30V normally aspirated: Coolant after run pump?

[SuffolkD at aol.com]

Strange battery swap finding.

The ~ 560 CCA OEM battery wouldn't take a charge after 7 years.

When I'd turn the key, it would go click, click, click even with a charger on 
it........

I noticed a trickle in the coolant expansion tank after starting attempts ( 
no engine turn over, just clicks.)

Is there an after run circuit (pump) on the 99 A6 30V 6 cyl?

-TIA -Scott by BOSTON
no safe mode on battery either, I kept juice to the terminals during the 
switch.

Autozone $59 bux 700 CCA. BTW, Consumer Reports rated it in the top 
tier........