Look at the leftmost screenshot you posted. See the little icon just to the right of the search bar, roughly in the middle of the screenshot? Click it to change display settings.

I can see the little icon to the right of the search button. But the icon menu don't display all columns, only a few columns. There are more columns in the settings of display columns button in thunderbird old versions. And i can't select columns place order.

But it does exactly what you said you wanted, the ability to show name, email, home and cell phone.

Thunderbird 102.3.1 appears to add more options to that same dropdown, so maybe more are on the way, but... it doesn't actually work!

I've long since used the "Organization" field to know when people joined and quit a club I run. I can now add it, but unless I edit an entry in one of my address books, nothing appears. In the second screenshot below, you can see ONE address with the field populated, but it's only because I opened and saved that one. Every other address in that address book also contains text in that field.

So unfortunately, if you use this feature, it appears that you need to open and save every single entry for it to work. :(