Recovering data from the HD of a dead Mac

Remove the drive from the old computer and use a bare-drive adapter to connect it via USB to the new computer (I'm assuming the new one is also an iMac with USB ports--you didn't say).

This is a bare-drive adapter:

https://eshop.macsales.com/item/NewerTech/U3NV2SPATA/

It includes a power supply to spin up the old drive.

Great! As you already know how to take it apart, you have 2/3rds of the battle won!

Here's a tip in case you have to live through this again: I was in the same boat when my 2010 iMac died suddenly in 2018. I didn't have to extract the old drive! Several years before I'd added a second external drive to the 2010 used exclusively for bootable backups made with Carbon Copy Cloner.

It proved far easier to use the newest CCC back to recover files than with Time Machine. I don't like Migration Assistant and so set up new computers by transferring only my data files. None were lost.

You don't need a huge drive because you can tell CCC how long to keep backup. I've set mine for a generous 45 days and it is using less than 500GB of a 1TB external. Set to 14 days and I bet you could get by with a 256GB drive.

That won't help with the dead one but it is something to consider with the new one. The nice thing is this method can recover data without involving suction cups and a screwdriver!

See if you can start the iMac up in Target Disk Mode.  If you can then you can connect it to another Mac with an Ethernet cable and access the hard drive from the other Mac.