I'm still slightly confused because you again mixed up iPhoto and Photos when you said:
At this point, a window opens saying "Cannot import items. These items cannot be added to your library because file format is not supported by Photos".
and then
So next, I go back< to Pictures, then choose>the "Photos Library" folder. Having clicked on this folder, the iPhoto application opens and says a repair is being performed (I assume the repair is being done to the photos on the external HD, to make them compatible with the new version of iPhoto??).
The old photos are then accessible in iPhoto. But with a few photos missing... which is the part which most mystifies me!
What's confusing is are you running
Photos or
iPhoto? You still mention both.
For education, when you look at the TM backup in Finder, as you said you did, the "files" you see there are NOT files, but symbolic links to the files. That's how TM works. When you first backup your system everything is literally copied over to the backup drive. At the next backup, TM looks for those files which have changed, and copies ONLY those over, but then creates symbolic links in the new backup the the files in the original backup. At the third backup, it repeats that process. At that point the backup has some original files with two symbolic links to them chained together, some files that changed in the first backup with symbolic links in the third backup and some files that are in the original backup with symbolic links in the second, but a new version in the third. And each backup has other NEW files that were first created in either the second or third backup. Confused? Look at these three lines:
A B C <-- original backup with files A, B and C
A' B' C+ <-- first backup (the letters with the "'" are not files, but links to the original A files, C+ is a new version of C)
A+ B' C+' <-- Now we have a new A version, but a link to the not-so-new C+ file. The B file is unchanged from the first backup.
This use of links continues through all of the TM backups. What that means is that although Finder shows a file, what is really there may be the head of a very long chain of symbolic links all the way back to the original backup. And when Photos tries to import what you point to, if it's a link, it may not work because it's not a real file, but a symbolic link. I don't know that for certain, but given the messages you see, that makes some sense.
Let me ask, have you tried Migration Assistant to migrate from the TM Backup to the new laptop? Not TM, but Migration Assistant? I wonder if MA could get the old iPhotos library imported for you from the TM backup, and from there import to Photos. MA can read TM backups (assuming your poking around hasn't destroyed the integrity of the backup linkages).