The point is that the state no longer cares if it's 'holy matrimony' (copyright Christianity-de-jour) or 'Allah's Blessing' or whatever the Judaist equivalent might be - they're all 'blessings', and therefore for the state's purposes entirely irrelevant.
Correct - marriage has always been a legal construct within particular societies and without the appropriate legal element there is, frankly, no marriage.
VG's link is clear on this as it describes situations where individuals have engaged in a religious "marriage" ceremony but failed to complete the legal requirements for marriage and the position is clear - this is a "non-marriage". In other words these people are not married regardless of what any religion may claim.
So religions can install all sorts of "marriage" ceremonies etc, but without the legal part these are really nothing more than a blessing or a ceremony - these aren't marriages and the couple isn't married without the legal part.
So I guess the question here is firstly whether we should keep the legal part and the religious ceremony separate - as is already the case for most religions as their religious officials aren't registrars and their premises aren't registered. That seems to be Outriders preference and this would bring, largely CofE and RCC in line with other religions. Alternatively we can allow the legal bit (the actual marriage) to be embedded within the religious bit (the "non marriage") but surely if we want equality we should support this for all religions, not just a few as seems currently to be the case with CofE, in particularly having special privileges in this regard.
Unlike Outrider I think I'd go for the latter approach as personally I think we should allow people to get married (legally married) in as many places as possible, as this will often be very personal to the individual. So we once had a situation where your choice was a pretty church (which would require a religious ceremony) or a austere civic building for a civil ceremony. We've moved on, allowing all sorts of beautiful, but non religious, venues to be licensed for marriage. I don't really see an issue with this being extended to other religious buildings. Indeed I think this would reduce the likelihood of non marriages where couples have gone through a religious "marriage" ceremony but aren't actually married.