Syncing contacts to multiple places
There's some things that still aren't that easy to do on an iPhone/iPad and require the use of a Mac (or PC, however steps are harder) to accomplish.
Two parts to your question - first off going into Contacts--&gt; Groups is simply a way to turn on and off certain groups of contacts so you are only seeing the groups you want to see. In your situation you would want to have the checkmarks ON next to all the Groups, otherwise you will see the behavior you are describing (accidentally hiding exchange contacts).
Next, in order to get the Exchange contacts migrated over to iCloud the easiest method is to get ahold of a Mac and follow these steps:

(Optional) If this is not your computer, or it's shared with others, consider creating a fresh user account for your Wife in order to do this task so it doesn't mix up content with other people. To do so Click Apple menu in upper left and go to System Preferences. Then click Users &amp; Groups. Click the lock symbol in the bottom left and unlock the window pane with the computer's password. Then click the + sign in the bottom left and create a new user account for your wife to use for this task. Sign out of the account and into her account -- skip through the setup assistant the best you can, but do sign into her iCloud account.
Go to System Preferences, then click on Internet Accounts.
Add an exchange account and sign in with your wife's info. Then enable the info you are hoping to transfer - in this case just the Contacts.
If you didn't do it in the setup assistant, add your wife's iCloud account and make sure to enable Contacts.
Open the Contacts application on the dock and look to see if both her iCloud contacts and her Exchange contacts show up ok.
If they do, click on the Exchange section where it says &quot;All Exchange&quot;. Select all (Command, A on the keyboard) or edit menu, select all. Then drag them onto the iCloud section and be patient as it copies over.

Once you've copied over the Exchange contacts to iCloud, you can then go back to the iPhone and enable the contacts for both Exchange and iCloud in the Settings--&gt; Passwords &amp; Accounts section. Or if you're running iOS 14 or newer it would be in Settings--&gt; Contacts, then click Accounts near the top and make sure Contacts is enabled on both Exchange and iCloud.
(Optional) Last tip that helps a lot - in the same Contacts settings page as mentioned above, you can look for the option &quot;Default Account&quot;. Once you find it set her default account for contacts to iCloud. Doing that will help her out a lot in the future as new contacts will go on her iCloud.
I love Exchange but have never felt comfortable putting my personal contacts onto a workplace email account that I could lose access to on a moments notice - hence my recommendation for her to set the default contacts to iCloud.

Regarding setup if you don't have a Mac, you can use a newer browser like Chrome/Firefox/Edge, and sign into iCloud.
Select the Contacts section and in the bottom left is a cog that will give you the option to &quot;Import Vcard&quot;.
Next is the convoluted process of exporting the contacts as a multiple contacts vCard which basically involves exporting the contacts as a CSV file out of Outlook, then using Gmail's import process, then exporting them back out as a Vcard for Apple products.
Here's reference for those steps: https://www.msoutlook.info/question/803
Afterwards you would get that long sought after group Vcard you created with the steps mentioned above and upload it into iCloud.
Lastly jump back up to my steps mentioned after 6. and you'll be all set from there.