Manual migration

You do not want to perform a manual migration.

Have a last Time Machine backup from the 2012 Mac mounted on the new iMac when you first power it up. There will be an initial setup process and within it will be a question of migrating data from a Time Machine drive. You want to select that, and it will take a moment to find your TM drive, and then begin transferring third-party applications and all of your personal data, email, and other preference settings to the new machine.

When setup is done, and immediately after the reboot, remove your Catalina TM drive and keep it in a safe place. Do not resuse it with Monterey, and do acquire another drive that is 2.5 - 3x larger than the new iMac's drive. The new 7200 rpm rotational, or SSD TM drive must be reformatted as APFS in Disk Utility before its first Monterey TM backup.

When you reboot, you should be greeted with your Catalina configurations on Monterey with minimal further tweaks. I always use this process and am quite satisfied with it. I never use the migration assistant over the network between Macs as that is an infinity joke. The name of your iMacs (system preferences > Sharing panel) must be unique, just as their network address.

I agree with VikingOSX. Don't do what you are proposing.

Create a Time Machine backup: Back up your Mac with Time Machine - Apple Support. When you get the new Mac, part of its initial setup asks if you want to transfer information from another Mac or its Time Machine backup. Do that.

Unlike VikingOSX I have never encountered a problem with migration over a network, but I recommend using Ethernet whenever practicable. Migrating over Wi-Fi invites the unpredictable vagaries of one's local wireless environment.

This describes what I have done with multiple Macs over the course of decades.

But I’m worried there might be something I’m not considering.

Such as the two replies you received?

Get the new Mac and move things across the way you described. Find out what works and what doesn't, and then you will be the expert.

Since you asked though,

1). The only existing program I really care about is Adobe Lightroom (My wife is a photographer and this is her machine). Is there anything in the preferences I need to move?

Move everything.

2) ... Do you agree that is the best idea?

No. It's not even a good idea.

3). Is there anything else I’m not thinking of which I need to copy over from the old machine, such as stuff in the system folder or preferences?

Move everything.

Let us know how it goes. You can always start over and do what I recommend though, so you have nothing to lose.

I am not offended. I am motivated to help you, and that's why I am suggesting what I did. Apple designed Macs to work for people like you. Not computer experts. What you proposed is complex and error-prone; a project something computer experts might enjoy.

The Genius at the Apple store could not give me any reason for the machine’s other than the fact that it’s old.

Unless you want to pay someone for a thorough diagnosis and then pay more for what is almost certain to be a cost-prohibitive repair, that's what you can expect an Apple Genius to say. Macs don't fail just because they're old; they have spinning hard disk drives that wear out and need to be replaced. Ten years is about the best you can expect from them.

It is premature to conclude that is what's going on with yours. We can perform some basic troubleshooting to confirm it if you want to, but the symptoms you describe are consistent with disk failure. Replacing an iMac hard disk drive would cost you a minimum of $400, and Apple won't repair that Mac anyway. It's too old. That's why they said what they said: buy a new one.

It's not my intent to discourage you from doing what you proposed either. I sincerely meant what I wrote: try it. I just think it's a waste of time.

I was alluding to how long (in my experience) that Migration Assistant takes over Ethernet between two Macs located two feet apart. The Time Machine drive solution is much faster for me.

A directly connected TM backup drive will certainly be much faster than any network connection. Maybe ten times faster.

Especially since I use Crucial X8 1050 MB/s TM drives…