Hi there, Bob593251! Thanks for reaching out to Apple Support Communities.
After reviewing your post, it sounds like you are looking to save a website on your iPhone home page. We definitely understand that would handy for a frequently visited site and we'd love to help. The instructions you are following sound correct. The path to save a website as an icon on your iOS device Home Screen are to open the website in Safari, tap Share, scroll down to Add to Home Screen, then tap Add after naming the icon.
If you are not seeing that icon, it may be that you are using a different browser other than Safari. The site you are trying to save also may not support that. Check with another website to confirm. Also, this may simply be a bit of unexpected behavior, which can be cleared up by restarting your iPhone: Restart your iPhone.
You may also wish to save that website in your Safari bookmarks, which will also allow you fast access to it: Bookmark favorite webpages in Safari on iPhone.
Please let us know if you have follow-up questions about this issue, or if the issue remains unresolved. Have a great day!