Good grief, I don’t even remember writing this OP, the ties obviously didn’t bind me very tightly to the topic.
I still protest at it’s placing in Cafe Society, however.
Good point.
The phrase “the tie that binds” instantly makes me think of the hymn, and I’ve always assumed that’s the origin of the phrase, and that other uses of it are direct or indirect allusions to that hymn. But I don’t know for sure.
Footnoting the well(?)-known parody verse:
Blessed be the tie that binds
My collar to my shirt.
I’m wasting no dollars,
I’m buying new collars
To hide that ring of dirt.
The ties that bind us together are the common experience, the shared values, our beliefs, etc. that make our relationship with each other stronger. It’s not about bondage, slavery, etc.
Springsteen has a song the Ties that Bind from his 1980 album the River
Generally speaking, the “tie” is a commonality that bonds us together.
In the hymn, specifically, the “tie” is Jesus Christ that binds all Christians in a bond of love.
I think it’s only because the hymn has a clumsy tune, or clumsy lyrics, that this sentence ever became “a thing”.
The first words are “Blest be the tie that binds / Our hearts in Christian love” - but there’s a longer note on “binds”, and when inexperienced or uneducated singers find a longer note at the end of a line, they tend to pause, even if pausing messes up the meaning of the words.
And if there’s a short note at a logical stopping spot, they probably won’t stop either, leading to “The Lord’s my shepherd I’ll not want”, having a quick note on “shepherd”, sounding like “I’m not going to want this shepherd, but I guess I’m going to have one anyway”. This song has them all over - probably the worst is the long note on “lie”, so that “He makes me down to lie in pastures green, he leadeth me the quiet waters by” ends up meaning “He makes me lie down for no reason. He leads me in green pastures. The quiet waters. Bye!” 
So, “The tie that binds.” comes from church singers not paying attention. IMO.
That, and in my experience it’s typical for a hymnal to identify hymns by their first lines (rather than by some other title that isn’t just the first few words you sing).
True, though there is that 1970s-Catholic gem usually known as “On Eagles’ Wings” rather than “Yoo-hoo!” 
This thread is instructive.
I always thought it was a negative thing. The “ties” - marriage, family, friends - can “bind” us - reduce freedom of movement, or freedom of choice.
Like how you’re always stuck bailing your no-good brother out of jail, even though you’d rather let him rot.
That’s how I always understood it.
I think people long ago started borrowing the hymn words and using them ironically/sardonically/whatever term is right.