24" Intel iMac (2006) date and time keeps resetting on boot, even after new battery and NVRAM reset

Since a week my iMac, running Snow Leopard, behaves strangely in two ways:

First, I always unplug the iMac when I don't use it. Now, when I plug it in, it immediately starts, without me pressing the On-button.

Second, after booting, the date and time are 1st of January, 2001, midnight. Several programs complain instantly. The IP address is something strange, certainly not the usual 192.168.X.Y address. As internet access, I use a network cable connected to a Devolo home-plug (connected to a fibre glass modem).

After two minutes of waiting, the iMac gets a normal IP address. One minute later, the date and time are what they should be.

First, when I googled the symptoms, the cause seemed easy: just replace the internal CR2032 battery. After all, it's 14 years old. I did that and it didn't help. It did change the volume of the startup chime though, it's much louder now.

Then I reset the NVRAM/PRAM, with the four button thing when booting. That helped for one boot, but on the second boot, 1st of January 2001.

I booted in Safe mode. Several times resetting NVRAM/PRAM. I checked the system settings, but they are what they have been the last 10 years.

Unfortunately, nothing seems to work. Is there anything else wrong with the machine? Is there some other battery that needs to be replaced?

Sometimes I booted the iMac without the Devolo, and used WiFi. Even when it did not have an internet connection, the time was always what it should be. So the strange IP address should not be the cause of the iMac not knowing the right date/time.

Because I had 1% uncertainty about how I had placed the new battery in its holder and because I wanted to make sure it was firmly in place, I reopened the iMac. And what did I behold? The battery holder is supposed to be connected to the motherboard on two places, the upper and bottom side. And the upper side was loose! This must have happened while prying out the old battery, that is an even trickier operation than I thought it was.

So that was the cause. Now I have the challenge of fixing this. Duct tape (to hold the battery holder down on the board) didn't help, I can tell you that by now. It should be soldered, but I have neither the equipment nor the skill to do that :-(