Interesting results: I tried to add my passport to Apple Wallet last week, and it's been in the process of "verifying" for days now. In the meantime, I've been trying to add my Illinois driver's license. My phone (iPhone 17 Pro), and the Apple Wallet would not show Illinois as an option. I tried all the suggestions listed in this thread, up to the point of factory reset, which isn't worth the trouble. This morning, on a whim, I canceled the ongoing verification process for adding my passport, and.... voila! Illinois now shows up in Apple Wallet as a driver's license option.
So make sure you don't have an ongoing digital ID verification process running before adding your Illinois driver's license to your Apple Wallet.
FYI - I stopped the passport verification process by using the "Don't add to Wallet" option.