Water pump runs all the time

Something in our 2011 Leaf was draining the 12 volt battery, and the EV system amber warning light would come on (and stay on) after the car was driven a few hundred yards. Although the dealer service department checked it out and said that water pump #1 was running 24/7 and draining the 12-volt battery, what we found was that it was running any time the ignition was on or the car was being charged but not when it was just parked. They recommended replacing water pump #1 (at a cost of $985), and then checking the car again to see if that remedied the problem. They also suggested that what was currently happening could cause major damage to the car if not fixed. We are currently trickle charging the 12 volt battery when the car is in the garage. My question: Is there really any harm in continuing to drive the car "as-is" and maintaining the 12 volt charge with the trickle charger? If the pump is running more than it should be, it will still keep the electronics cool. Also, if the pump is not running when the car is just parked, why should it drain the 12 volt battery? I'm more than a bit skeptical about the information that the dealer is giving us.

As far as I can tell, my car has always run the water pump when actively charging or when on.

I'm more willing to believe it's just time for a new 12V battery, the LEAF is known to leave it undercharged which shortens it's life.

Whatever you do, don't let that dealer work on your car. He's trying to separate you from your hard-earned money. How old is your 12V battery? If I were you, I'd buy a new 12V battery (under $100). Chances are that will solve your problem. If not, use the trickle charger.

Agree with drees.
Nothing wrong with the water pump.
It is doing what it is supposed to do.

Dealer is incompetent.
You need a new 12V battery.
Amazing it didn't fail sooner.
I'm on my second warranty / pro-rated replacement.

Start trickle charging it often.
The new one will last longer.

Thank you everyone for the input! Sorry I neglected to mention in my original post that the car's original battery failed in June. The one that just failed was the new 5-year battery and had only been in the car for 2 1/2 months. This seemed to suggest to me that the car was doing something different to the battery since the original one had lasted over 4 1/2 years. It's only been in the last two weeks (since the installation of another new 12-volt battery) that the EV system warning light has been showing amber.

^^^^ It is possible for a relatively brand new 12V battery to have an infant failure.
All manufactured devices have some early failures. See Weibull curve (might have the spelling wrong).
I still think dealer or a local auto parts store should hook up machine and test the battery capacity.

Also possible that they messed up something when they swapped the battery.
Not very likely but possible.

Possible it was never charged fully and your driving pattern hasn't charged it fully yet.
LEAF charging algorithm never really keeps the 12V fully charged anyway.
Put it on a trickle charger for a long time till it indicates fully charged.
It won't hurt anything and might solve the problem.