Have not Hymns and (the more modern) 'Religious song' always been written for a target group or occasion?
Office Hymns written for the Divine Office, Processional Hymns written for special processions etc.
O.K hymns written for one purpose, say the Vexilla Regis have found use as Office Hymns. Various Office Hymns are edited excerpts of longer hymns.
Of course more recently we have quite a few songs written for one purpose used for a different one, but this is due to changes in ideas of worship and taste (or lack of).
What would Wesley say about his compositions being sung during Mass in a Catholic church? Or Faber, Caswall, Newman's hymns / songs replacing Propers at Mass?

What is the "tradition of hymn-singing" and how has it been broken?

The choir lofts we sing in are full of copies of the Westminster Hymn book (revised ed.), when was the last time we heard these hymn / songs being sung? What about the unrevised edition, are any of those still sung? These were popular and well known books and the offerings are unknown in most Catholic churches, and barely remembered in more traditional parishes.

When I was a child we had "The Parish Hymn book", this was perhaps the first of the edited Hymn / song books, a few hymns had lost verses, found in earlier books. This book was published in the 60's / 70's mainly for use outside Mass, but with the creeping novelty of the Offertory hymn (song) and Communion hymn (song). While a few of the songs therein will be recognised in a modern parish, the word would not be having been edited for content. I remember the first three songs from your list as a child, but I have not heard or sung these for many many years.

From the late 60's it has been popular to sing a ever varying collection of songs during Mass, this in my opinion cannot be described a Tradition. Up until the late 70's early 80's it was popular to sing another varying collection of songs and hymns, during Benediction, and other popular devotions, this could be described as a tradition as it had been going for more than a handful of years. But where are these popular devotions in the modern church?

We live in an evermore inwardly focused Church, that has ever greater number of services tailored for small groups, children's Mass, the youth Mass, Spanish Mass etc. There is no real popular need for a core repertoire of songs or hymns, this is shown by the never ending new editions of the latest song / hymn books.

BUT you are quite right there should be a core set of songs / hymns that we sing together, both young and old, and from the 4 corners of the world. Much to my father's never ending annoyance his grandchildren (my children) are not learning the songs / hymns he fondly remembers as a child, nor are they learning the songs I remember. They are instead learning the Hymns and Antiphons sung by now long gone generations, yes they are learning truly traditional church music sometimes known as Gregorian Chant.