I can’t replicate what you describe - web links in text pasted from the web into a note then copying that text to the text of another note. When i do that the web link survives.
Tinderbox links are described here. ‘text links’, links created from anchor text in a note’s text (otherwise referred to as its $Text attribute), are not persisted during a paste. The design reason for this is not known. You would need to ask Tech Support - be aware this is a user-to-user forum, so you’re talking with fellow users here (disclaimer: I am a forum admin and not an Eastgate employee).
The reason I can’t replicate the problem in your note title is the rich text engine in the $text area will detect (web) links in pasted material and create functional ‘smart links’ using routines based into the Apple frameworks. These links exist only in the RTF, i.e. are not internal Tinderbox, and will paste between notes.
So although your note title describes the latter, your question actually relates to the former type of link. ‘smart links’ are fairly new and the fact they aren’t adopted as Tinderbox likely boils down to some unseen complexities. Tinderbox isn’t an outlining utility but rather a toolbox for notes. This means it doesn’t—like an outliner—just do outlines. There is generally more than one way to do things, allowing for different styles of work, and no expectation that most users will use more than some of the features on offer.