Could iOS disallow apps from looking at the clipboard?

If you want to keep the cut-n-paste functionality, programs will at some point eventually have to read the clipboard contents.

The operating system (on iOS or macOS) in general cannot know what the "paste action" actually is. In some programs it could be clicking/tapping an icon, it could be pressing a keyboard combination (not necessarily the standard one), or it could be programmatically triggered.

However the operating system could trigger a popup asking the user whether to allow a paste operation, when the app accesses the clipboard contents. Or it could show some other indication that the clipboard is being accessed. This is what happens in iOS 14 with a small notification that the clipboard contents was copied.

Note that the operating system cannot as such know whether this popup is being triggered after the user pressed/clicked "Paste", or there is some other reason!

There are other reasons for accessing the clipboard than purely "paste" operations. Some apps for example detect that you have in the clipboard and display certain actions for the user. For example if you have a link in your clipboard, and you start up a link manager app, it could offer easy access to add that link into its database.

Another example is the Facebook app that back in 2015 introduced an automatic "share" suggestion when you have a link in your clipboard.

In iOS 14, Apple has added functionality to allow app developers to ask if the clipboard contains a link that matches a certain pattern without actually accessing the clipboard contents and triggering the popup. This could allow apps to keep their helpful clipboard features without the same privacy impact.

Could iOS disallow apps from looking at the clipboard?

No, it can't. Since iOS 14, it can display a warning.

Is there any reason for the operating system to allow programs to view the clipboard not in response to the paste command?

Yes in an ideal world where any application would only do legal things, and where a phone or computer was used by a unique user within the same work context (professional, private) it would help us to save time.

In any other case, this could easily lead to privacy violation or company data leak or criminal data theft.

For example, after you copied an image, if you open Preview on a Mac, and it will read your clipboard and see you have pasted an image and offer you to create a new "Untitled" document from Clipboard. This function was conceived in an ideal world (the one where we don't need keys for car or house).

If this image is an unclassified document in an ideal world: no problem. Otherwise, many potential problem with some of them with consequences which could be prosecuted (confidential information leak).