Why can’t I open my new apps and why do they not show up on my Home Screen

What do you mean you can't open your new apps? Are you saying that you can't open them just because you can't find them?

If you are using iOS 14, they might download to the App Library and not show on the Home Screen.

If that's the case, you can move them out to your Home Screen

From your Home Screen, swipe left until you get to the App Library.

Swipe downward and you'll get an alphabetical list of your apps.

Locate the app in the list or type the app name into the search field

Once you've located the app, tap/hold until it raises/selects, then drag it and it'll pop out to your Home Screen

Organize the Home Screen and App Library on your iPhone - Apple Support

If you want apps to be added to the Home Screen when downloading,

check Settings > Home Screen and place the checkmark by Add to Home Screen