In this forum, the word 'liberal' is normally used in a deprecative manner. Very few here would describe themselves as Anglo-Catholics as that term was really applied to the Oxford Movement and those who followed it. It is certainly not a term I would use, though I am happy to be seen as a Catholic Anglican. If you were to ask me for my preferred descriptor, I might use the term Anglidox. I am however very aware that I have a higher view of the 39 Articles and the Chicago-Lambeth Quadrilateral than many AnglicansI know who would run a mile from anything Catholic or Orthodox, preferring to describe themselves as Christian, Protestant, Born-Again Bible-Believing ... What I am saying is that the labels don't really work all that well.
This sub-strata has been at times called 'affirming Catholics'. ++Robert Duncan would be an example. Anglo-Catholic, but championing women's ordination, but repulsed by gays - quite a spectrum of positions.
It depends on where they’re coming from, but I think for some part of the attraction of Anglo-Catholicism is getting away from Fundamentalism. So if one’s “Catholicism” does not entail obedience to Rome, it’s easy to see how endorsement of Tradition in a way that’s more fluid might dovetail with more traditionally “liberal” commitments.
If you look at Church of England 2001–2019 the Cof E appears to have been in decline for decades with constant linear rather than constant percentage decline. Based on these data it is forecast that extinction will be reached around 2060. I doubt linear decline will continue until extinction but the CofE will be left as an insignificant and irrelevant rump. The data indicates that progressive churches are in decline with traditional/evangelical churches growing. I have heard Youtube commentators (e.g. Mike Winger) say that generally churches that adopt progressive theology soon decline dramatically but haven't until now seen actual data supporting that observation. This looks like the theological equivalent of Go Woke, Go Broke.
The folks the OP has described are sometimes called Affirming Catholics. And much as the Romans might hate what I'm going to say, their average parishioner fits the mold as well as any Episcopalian.
And continuing the theme I just heard that Southern Baptists Lost Nearly Half a Million Members Last Year. The article attributes some of this to updating church rolls and notes that baptisms increased. However that increase is still in line with the decline experienced this century, in fact they've barely reached the declining trend line.
I think there are a number of things to bear in mind. Firstly if one part of the church hurts, we all hurt. It is worth remembering that Christianity does not have a plural. There may well be multiple expressions of Christianity, but there is only one Christianity. We affirm this every time we say the Nicene Creed and announce the four notes of the Church - One - Holy - Catholic - Apostolic. The issues that confront the SBC will undoubtedly have parallels to other Churches experiencing decline, however, they will have a number of their own problems. I don't understand them to be theologically progressive, or liberal, and my general understanding is that they are not LGBTQIA... affirming or friendly. Many Churches that present absolutes and often very authoritarian are good at attracting new members, as many people in our world today crave certainty. Many of these Churches have big back doors and become places where people camp for a time on their journey. There are many fewer nominal Anglicans than there used to be, and many fewer cultural Anglicans. I have a feeling that where Anglicans stay it is because faith is significant. I suspect that being successful is not as important as being faithful.
I agree with both your posts on this thread (thus far). Describing oneself as an Anglo-Catholic (although we prefer the term 'Catholic') once meant one thing. As it always seems to be in Anglicanism that branch of the Anglican Tradition now encompasses a variety of views. I think it would be correct to say those who say they are Anglo-Catholic and espouse so-called 'liberal' views are those who would identifiy as Affirming Catholics.
It's ironic because historically Tractarianism and Anglo-Catholicism developed as a critique of theological liberalism and associated secular changes in British society in the 1830s and 1840s.
The Church of England, like other denominations, has for some time been losing numbers in its membership. Being the national church it, also like other denominations, had its share of 'nominal only' members, which to all intents and purposes, apart from swelling the membership total, contributed NOTHING WHATEVER or at best very little, to the Christian mission of the Church of Christ on earth. Nominality (falling away from faith), has existed in the church since it first began to be the church, so nothing new there. We worry perhaps too much about 'decline' and even more too much about the church becoming more aware of and vocal about injustice and oppression, which in the past it has sometimes even itself participated. The Anglican church is finally waking up after a two or three century sleep during which it had become almost merely an adjunct to national asperations to colonialism, in the name of supposed 'enlightenment' of the natives. It is no longer seen as the natural expression of National Identity that the King James translation and the Book of Common Prayer were originally intended to encourage among a sovereigns subjects. From what we can read in scripture of the church in Corinth, we can assume that 'woke'-ness (in its pejorative sense), in the church was there from the beginning. For all THAT Paul did not accuse it of NO LONGER BEING the church. He just tried to put right what was wrong with it, (which seems to have been quite a lot more, in his opinion, than just some women talking in church). 1 Corinthians is a very good way of examining what church can be like and what Paul thought could be done to improve it. Paul, notice, never suggested their faults though would inevitably result in a decline in numbers. Even if he thought it might, he wasn't concerned about that. True, when a church finally begins to wake up to its own imperfection and address its prejudices and hypocrisies, i.e. becomes 'woken' from complacent, self satisfied, slumber, it will be groggy and bleary eyed, making mistakes and not being yet fully alert to its duty in Christ's purposes but at least it will at last be becoming 'awoke'. .
I don't have that much to add, coming late to the party, but growing up I was disillusioned to find that at least one Episcopal priest I considered "high," wearing Eucharistic vestments, crossing himself, and putting Christus Rex crucifixes in the church, wasn't in fact conservative. Not a classic Anglo-Catholic: I thought A-Cs were basically pre-Vatican II Romans with thous and thees; I dare say many Continuers still do. Anyway, confusing! Liberal A-C's become the mode in that church. As I say, the Church of England is different; what I call Episcopalians (liberal high church) vs. Evangelicals. And where do you draw the line between high and Anglo-Catholic? After Porvoo, do the Anglicans still say the episcopate is necessary? The brass'n'class of British civic religion, royal ceremonies in church or churchmen at civil ceremonies, is high but not exactly Catholic. It seems to me an A-C believes in a church with necessary episcopal authority ("high church" originally meant that, not ceremonial!), even the church infallibility I believe in, and pretty much transubstantiation even if they don't call it that about the Eucharist. I once met a PCA Presbyterian gentleman who thought Christians should unite and divide based on their views on the Eucharist, so the ancient high churches such as Orthodox and Catholic would be one church, classic Lutherans and classic Anglicans another, and Reformed yet another. It may not work - Affirming Catholics (as they used to be called) hold a high enough view of the Eucharist. But a nice idea. In the 1580s I would have been a Church Papist like Byrd.